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Living Rev. Relativity, 6, (2003), 2 http://www.livingreviews.org/lrr-2003-2 Gravitational Waves from Gravitational Collapse Chris L. Fryer Los Alamos National Laboratory MS B227, T-6 Los Alamos, NM 87545 U.S.A. email: [email protected] Kimberly C. B. New Los Alamos National Laboratory MS T085, X-2 Los Alamos, NM 87545 U.S.A. email: [email protected] Accepted on 13 January 2003 Published on 10 March 2003 Living Reviews in Relativity Published by the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) Am M¨ uhlenberg 1, 14424 Golm, Germany ISSN 1433-8351 Abstract Gravitational wave emission from stellar collapse has been studied for more than three decades. Current state-of-the-art numerical investigations of collapse include those that use progenitors with more realistic angular momentum profiles, properly treat microphysics issues, account for general relativity, and examine non-axisymmetric effects in three dimensions. Such simulations predict that gravitational waves from various phenomena associated with gravita- tional collapse could be detectable with ground-based and space-based interferometric obser- vatories. This review covers the entire range of stellar collapse sources of gravitational waves: from the accretion induced collapse of a white dwarf through the collapse down to neutron stars or black holes of massive stars to the collapse of supermassive stars. This 2006 revision was written by Chris L. Fryer and Kim New jointly. c Max Planck Society and the authors. Further information on copyright is given at http://relativity.livingreviews.org/About/copyright.html For permission to reproduce the article please contact [email protected].

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Page 1: Gravitational Waves from Gravitational Collapse · Gravitational Waves from Gravitational Collapse 9 2 Accretion Induced Collapse 2.1 Collapse scenario Stars with masses below 8M

Living Rev. Relativity, 6, (2003), 2http://www.livingreviews.org/lrr-2003-2

Gravitational Waves from Gravitational Collapse

Chris L. FryerLos Alamos National Laboratory

MS B227, T-6Los Alamos, NM 87545

U.S.A.email: [email protected]

Kimberly C. B. NewLos Alamos National Laboratory

MS T085, X-2Los Alamos, NM 87545

U.S.A.email: [email protected]

Accepted on 13 January 2003Published on 10 March 2003

Living Reviews in RelativityPublished by the

Max Planck Institute for Gravitational Physics(Albert Einstein Institute)

Am Muhlenberg 1, 14424 Golm, GermanyISSN 1433-8351

Abstract

Gravitational wave emission from stellar collapse has been studied for more than threedecades. Current state-of-the-art numerical investigations of collapse include those that useprogenitors with more realistic angular momentum profiles, properly treat microphysics issues,account for general relativity, and examine non-axisymmetric effects in three dimensions. Suchsimulations predict that gravitational waves from various phenomena associated with gravita-tional collapse could be detectable with ground-based and space-based interferometric obser-vatories. This review covers the entire range of stellar collapse sources of gravitational waves:from the accretion induced collapse of a white dwarf through the collapse down to neutronstars or black holes of massive stars to the collapse of supermassive stars.

This 2006 revision was written by Chris L. Fryer and Kim New jointly.

c©Max Planck Society and the authors.Further information on copyright is given at

http://relativity.livingreviews.org/About/copyright.html

For permission to reproduce the article please contact [email protected].

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How to cite this article

Owing to the fact that a Living Reviews article can evolve over time, we recommend to cite thearticle as follows:

Chris L. Fryer and Kimberly C. B. New,“Gravitational Waves from Gravitational Collapse”,

Living Rev. Relativity, 6, (2003), 2. [Online Article]: cited [<date>],http://www.livingreviews.org/lrr-2003-2

The date given as <date> then uniquely identifies the version of the article you are referring to.

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Article Revisions

Living Reviews supports two different ways to keep its articles up-to-date:

Fast-track revision A fast-track revision provides the author with the opportunity to add shortnotices of current research results, trends and developments, or important publications tothe article. A fast-track revision is refereed by the responsible subject editor. If an articlehas undergone a fast-track revision, a summary of changes will be listed here.

Major update A major update will include substantial changes and additions and is subject tofull external refereeing. It is published with a new publication number.

For detailed documentation of an article’s evolution, please refer always to the history documentof the article’s online version at http://www.livingreviews.org/lrr-2003-2.

24 Jan 2006: This submission is a revision to the “Gravitational Waves from GravitationalCollapse” review by Kim New - it was written by Chris L. Fryer and Kim New jointly. Thisrevision focused mostly on updating the past review and adding more movies. The major changesare as follows:

Page 7: Revised paragraph above and added references.

Page 9: Revised section.

Page 9: Revised section, added Figure 1 and reference.

Page 10: Revised section and added references.

Page 11: Added reference.

Page 16: Revised section and added references.

Page 18: Revised section.

Page 19: Nakamura’s 2D simulations moved to Section 4.3.

Page 22: Added reference.

Page 22: Added references.

Page 23: Added references.

Page 25: Extended paragraph above.

Page 25: Updated references.

Page 26: Added movie by Ou et al. (2004)

Page 27: Removed paragraph on Davies et al. (2002).

Page 27: Added reference.

Page 30: Shibata’s simulations moved to Section 4.3.

Page 30: Revised section and added Figures 12 and 13.

Page 32: Added references.

Page 35: Added movie.

Page 36: Revised paragraphs above and added Movie 17.

Page 36: Added reference.

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Page 37: We have taken pieces of the “Collapse of Massive Stars” section to develop a “Col-lapsar” section.

Page 40: Revised paragraph below and added reference.

Page 44: Added reference.

Page 45: Updated summary.

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Contents

1 Introduction 7

2 Accretion Induced Collapse 92.1 Collapse scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Formation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 GW emission mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Numerical predictions of GW emission . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Full collapse simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Analyses of equilibrium representations of collapse . . . . . . . . . . . . . . 14

2.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Collapse of Massive Stars 163.1 Collapse scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Formation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 GW emission mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Numerical predictions of GW emission . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.1 Historical investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2 Axisymmetric simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.3 Non-axisymmetric simulations . . . . . . . . . . . . . . . . . . . . . . . . . 233.4.4 General relativistic simulations . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.5 Simulations of convective instabilities . . . . . . . . . . . . . . . . . . . . . 30

3.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Collapsar Models 374.1 Collapse scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Formation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 GW emission mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Numerical Predictions of GW emission Mechanisms . . . . . . . . . . . . . . . . . . 384.5 Going further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Collapse of Population III Stars 405.1 Collapse scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2 Formation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 GW emission mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.4 Numerical predictions of GW emission . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Collapse of Supermassive Stars 426.1 Collapse scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2 Formation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.3 GW emission mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.4 Numerical predictions of GW emission . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 Summary 45

8 Acknowledgements 48

References 68

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Gravitational Waves from Gravitational Collapse 7

1 Introduction

The field of gravitational wave (GW) astronomy will soon become a reality. The first genera-tion of ground-based interferometric detectors (LIGO [45], VIRGO [126], GEO 600 [104], TAMA300 [182]) are beginning their search for GWs. Toward the end of this decade, two of these detectors(LIGO, VIRGO) will begin upgrades that should allow them to reach the sensitivities necessaryto regularly detect emissions from astrophysical sources. A space-based interferometric detector,LISA [181], could be launched in the early part of the next decade. One important class of sourcesfor these observatories is stellar gravitational collapse. This class covers an entire spectrum ofstellar masses, from the accretion induced collapse (AIC) of a white dwarf through the collapseof massive stars (M > 8 M) including the “collapsar” engine believed to power long-durationgamma-ray bursts [261], very massive Population III stars (M = 100 – 500M), and supermassivestars (SMSs, M > 106 M). Some of these collapses result in explosions (Type II, Ib/c supernovaeand hypernovae) and all leave behind neutron star or black hole remnants.

Strong GWs can be emitted during a gravitational collapse/explosion and, following the col-lapse, by the resulting compact remnant [244, 172, 173, 73, 214, 91, 88, 120]. GW emission duringthe collapse itself may result if the collapse or explosion involves aspherical bulk mass motion orconvection. Rotational or fragmentation instabilities encountered by the collapsing star will alsoproduce GWs. Asymmetric neutrino emission can also produce a strong gravitational wave signa-ture. Neutron star remnants of collapse may emit GWs due to the growth of rotational or r-modeinstabilities. Black hole remnants will also be sources of GWs if they experience accretion inducedringing or if the disks around the black hole develop instabilities. All of these phenomena havethe potential of being detected by gravitational wave observatories because they involve the rapidchange of dense matter distributions.

Observation of gravitational collapse by gravitational wave detectors will provide unique infor-mation, complementary to that derived from electromagnetic and neutrino detectors. Gravitationalradiation arises from the coherent superposition of mass motion, whereas electromagnetic emissionis produced by the incoherent superposition of radiation from electrons, atoms, and molecules.Thus, GWs carry different kinds of information than other types of radiation. Furthermore, elec-tromagnetic radiation interacts strongly with matter and thus gives a view of the collapse only fromlower density regions near the surface of the star, and it is weakened by absorption as it travelsto the detector. In contrast, gravitational waves can propagate from the innermost parts of thestellar core to detectors without attenuation by intervening matter. With their weak interactioncross-sections, neutrinos can probe the same region probed by GWs. But whereas neutrinos areextremely sensitive to details in the microphysics (equation of state and cross-sections), GWs aremost sensitive to physics driving the mass motions (e.g., rotation). Combined, the neutrino andthe GW signals can teach us much about the conditions in the collapsing core and ultimately thephysics that governs stellar collapse (e.g., [7, 87]).

The characteristics of the GW emission from gravitational collapse have been the subject ofmuch study. Core collapse supernovae, in particular, have been investigated as sources of gravita-tional radiation for more than three decades (see, e.g., [203, 245, 204, 57, 179, 171, 233, 74, 170,271, 198, 86, 88]). However, during this time research has produced estimates of GW strengththat vary over orders of magnitude. This is due to the complex nature of core collapse. Importanttheoretical and numerical issues include

• construction of accurate progenitor models, including realistic angular momentum distribu-tions,

• proper treatment of microphysics, including the use of realistic equations of state and neutrinotransport,

• simulation in three-dimensions to study non-axisymmetric effects,

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8 Chris L. Fryer and Kimberly C. B. New

• inclusion of general relativistic effects,

• inclusion of magnetic field effects, and

• study of the effect of an envelope on core behavior.

To date, collapse simulations generally include state-of-the-art treatments of only one or two ofthe above physics issues (often because of numerical constraints). For example, those studies thatinclude advanced microphysics have often been run with Newtonian gravity (and approximateevaluation of the GW emission; see Section 2.4). A 3D, general relativistic collapse simulation thatincludes all significant physics effects is not feasible at present. However, good progress has beenmade on the majority of the issues listed above; the more recent work will be reviewed in somedetail here.

The remainder of this article is structured as follows. Each category of gravitational collapsewill be discussed in a separate section (AIC in Section 2, collapse of massive stars in Section 3,collapsar models in Section 4, collapse of Population III stars in Section 5, and collapse of SMSsin Section 6). Each of these sections (2, 3, 4, 5, 6) is divided into subsection topics: collapsescenario, formation rate, GW emission mechanisms, and numerical predictions of GW emission.In the subsections on numerical predictions, the detectability of the GW emission from variousphenomena associated with collapse is examined. In particular, the predicted characteristics ofGW emission are compared to the sensitivities of LIGO (for sources with frequencies of 1 to104 Hz) and LISA (for sources with lower frequencies in the range of 10−4 to 1 Hz).

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Gravitational Waves from Gravitational Collapse 9

2 Accretion Induced Collapse

2.1 Collapse scenario

Stars with masses below 8 M end their lives ejecting their envelopes in a possible planetarynebula, leaving behind a white dwarf that gradually cools and fades away. For those white dwarfs inbinaries, binary accretion can reheat the white dwarf. If the accreted material ignites degenerately,the resultant nuclear explosion produces a nova and ejects all of the accreted material. But if thismaterial burns non-degenerately, the white dwarf will gain mass. When the mass of the whitedwarf exceeds the Chandrasekhar stability limit, it will begin to collapse.

Two possible fates await this collapsing white dwarf. If the collapsing core can achieve highenough temperatures, nuclear burning will begin and the stellar pressure will increase. This nuclearburning can drive nature’s largest nuclear bomb, producing type Ia supernovae. However, neutrinoemission from electron capture (e.g., URCA processes) can damp this burning until the core hascollapsed too deeply into its gravitational well for nuclear burning to turn around the collapse.The result of this collapse is the formation of a neutron star. Electron capture will dominate ifthe density at which nuclear ignition occurs exceeds a critical density ρcrit. For C-O white dwarfs,ρcrit is in the range 6 × 109 – 1010 g cm−3 [32]. For O-Ne-Mg white dwarfs, electron capture maybe stronger than nuclear burning under most conditions (if a Rayleigh-Taylor instability doesnot produce a turbulent burn front) [187, 127]. The collapse of an O-Ne-Mg white dwarf beginswhen its central density reaches 4 × 109 g cm−3. (For more details about the conditions underwhich AIC occurs, see [187, 127, 32, 31, 156].) The dynamics of the collapse itself are somewhatsimilar to the dynamics of core collapse SNe: the collapse proceeds until the core reaches nucleardensities, the core then bounces and sends out a bounce shock that stalls when it becomes opticallythin to neutrinos and loses its thermal energy. However, there is very little envelope around thiscore to prevent an explosion and the shock can easily be revived to drive a low-mass explosion.Exactly how much matter is ejected depends upon the details of the collapse calculation (compare[114, 263, 82]). Less than 10−1 M will likely be ejected by the star (due to the bounce itself ordue to neutrino absorption/wind mechanisms) [82].

2.2 Formation rate

The AIC occurrence rate is difficult to determine for a number of reasons. These include incompleteunderstanding of binary star evolution and determining how much matter is truly accreted onto thewhite dwarf versus the amount that is ejected through novae [48, 137, 234]. Another uncertaintyis whether the collapse of an accreting Chandrasekhar mass white dwarf results in a supernovaeexplosion or a complete AIC (with accompanying neutron star formation). Figure 1 shows oneestimate of the region in the space of initial white dwarf mass and accretion rate that producesAICs. New results continue to alter the dividing lines between these fates [234].

The AIC rate can be indirectly inferred from the observed amount of rare, neutron rich isotopespresent in the Galaxy. These isotopes (formed via electron capture) are present in the portion(∼ 0.1M) of the outer envelope ejected by the star during an AIC. The exact yield is sensitive tothe neutron fraction in this ejecta, which depends both on the neutrino transport and the electroncapture rates, but if all of these isotopes present in the Galaxy are assumed to have originated inAICs, an upper limit of ∼ 10−5 yr−1 can be set for the Galactic AIC rate [82].

Binary population synthesis analysis can be used to determine which accreting white dwarfswill undergo AIC. The results of Yungelson and Livio [268] predict that the galactic AIC rate isbetween 8 × 10−7 and 8 × 10−5 yr−1. Thus, a reasonable occurrence rate can be found for anobservation distance of 100 Mpc.

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10 Chris L. Fryer and Kimberly C. B. New

Figure 1: The final fate of accretion OMgNe white dwarfs as a function of the initial white dwarfmass and the accretion rate onto the white dwarf. (Figure 3 of [187]; used with permission.)

2.3 GW emission mechanisms

During AIC, emission of GWs will occur if the infall of matter is aspherical. The convective timeis likely to be brief, so it is unlikely that post-bounce convection will produce GWs. However,GWs will also be produced if the collapsing star or neutron star remnant develops rotational orpulsational instabilities [226, 227, 235, 252, 254]. These include global rotational mode, r-mode,f -mode, and fragmentation instabilities.

Global rotational instabilities in fluids arise from non-axisymmetric modes e±imφ, where m = 2is known as the “bar-mode” [239, 6]. It is convenient to parameterize a system’s susceptibilityto these modes by the stability parameter β = Trot/|W |. Here, Trot is the rotational kineticenergy and W is the gravitational potential energy. Dynamical rotational instabilities, drivenby Newtonian hydrodynamics and gravity, develop on the order of the rotation period of theobject. For the uniform-density, incompressible, uniformly rotating MacLaurin spheroids, thedynamical bar-mode instability sets in at βd ≈ 0.27. For differentially rotating fluids with apolytropic equation of state, numerical simulations have determined that the stability limit βd ≈0.27 is valid for initial angular momentum distributions that are similar to those of MacLaurinspheroids [221, 63, 162, 125, 192, 118, 248]. If the object has an off-center density maximum,βd could be as low as 0.10 [247, 260, 192, 49]. General relativity may enhance the dynamicalbar-mode instability by slightly reducing βd [223, 209]. Secular rotational instabilities are drivenby dissipative processes such as gravitational radiation reaction and viscosity. When this type ofinstability arises, it develops on the timescale of the relevant dissipative mechanism, which canbe much longer than the rotation period (e.g., [225]. The secular bar-mode instability limit forMacLaurin spheroids is βs ≈ 0.14.

In an attempt to reduce these high rotation requirements, there has been increasing workstudying bar-mode instabilities driven by dynamical sheer in differentially rotating neutron stars.Sheer instabilities excite the co-rotating f -mode. If viscous forces don’t damp this instabilityaltogether, it is possible that this instability can occur for βd-values as low as ∼ 0.01 for stars witha large degree of differential rotation.

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Gravitational Waves from Gravitational Collapse 11

In rotating stars, gravitational radiation reaction drives the r-modes toward unstable growth [5,77]. In hot, rapidly rotating neutron stars, this instability may not be suppressed by internal dis-sipative mechanisms (such as viscosity and magnetic fields) [152]. If not limited, the dimensionlessamplitude α of the dominant (m = 2) r-mode will grow to order unity within ten minutes of theformation of a neutron star rotating with a millisecond period. The emitted GWs carry awayangular momentum, and will cause the newly formed neutron star to spin down over time. Thespindown timescale and the strength of the GWs themselves are directly dependent on the maxi-mum value αmax to which the amplitude is allowed to grow [153, 154]. Originally, it was thoughtthat αmax ∼ 1. Later work indicated that αmax may be ≥ 3 [153, 236, 213, 154]. Some researchsuggests that magnetic fields, hyperon cooling, and hyperon bulk viscosity may limit the growth ofthe r-mode instability, even in nascent neutron stars [136, 135, 201, 202, 154, 151, 102, 6] (signif-icant uncertainties remain regarding the efficacy of these dissipative mechanisms). Furthermore,a study of a simple barotropic neutron star model by Arras et al. [9] suggests that multimodecouplings could limit αmax to values 1. If αmax is indeed 1 (see also [97]), GW emission fromr-modes in collapsed remnants is likely undetectable. For the sake of completeness, an analysisof GW emission from r-modes (which assumes αmax ∼ 1) is presented in the remainder of thispaper. However, because it is quite doubtful that αmax is sizeable, r-mode sources are omittedfrom figures comparing source strengths and detector sensitivities and from discussions of likelydetectable sources in the concluding section.

There is some numerical evidence that a collapsing star may fragment into two or more orbitingclumps [96]. If this does indeed occur, the orbiting fragments would be a strong GW source.

2.4 Numerical predictions of GW emission

2.4.1 Full collapse simulations

The accretion-induced collapse of white dwarfs has been simulated by a number of groups [14,166, 263, 82]. The majority of these simulations have been Newtonian and have focused on massejection and neutrino and γ-ray emission during the collapse and its aftermath (note that neglectingrelativistic effects likely introduces an error of order (v/c)2 ∼ 10% for the neutron star remnants ofAIC; see below). The most sophisticated are those carried out by Fryer et al. [82], as they includerealistic equations of state, neutrino transport, and rotating progenitors.

As a part of their general evaluation of upper limits to GW emission from gravitational collapse,Fryer, Holz, and Hughes (hereafter, FHH) [86] examined an AIC simulation (model 3) of Fryeret al. [82]. FHH used both numerical and analytical techniques to estimate the peak amplitudehpk, energy EGW, and frequency fGW of the gravitational radiation emitted during the collapsesimulations they studied.

For direct numerical computation of the GWs emitted in these simulations, FHH used thequadrupole approximation, valid for nearly Newtonian sources [169]. This approximation is stan-dardly used to compute the GW emission in Newtonian simulations. The reduced or tracelessquadrupole moment of the source can be expressed as

I-ij =∫

ρ

(xixj −

13δijr

2

)d3r, (1)

where i, j = 1, 2, 3 are spatial indices and r =√

x2 + y2 + z2 is the distance to the source. The twopolarizations of the gravitational wave field, h+ and h×, can be computed in terms of

...I- ij , where

an overdot indicates a time derivative d/dt. The energy EGW is a function of...I- ij . Equation (1)

can be used to calculate I-ij from the results of a numerical simulation by direct summation overthe computational grid. Numerical time derivatives of I-ij can then be taken to compute h+, h×,and EGW. However, successive application of numerical derivatives generally introduces artificial

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12 Chris L. Fryer and Kimberly C. B. New

noise. Methods for computing..I-ij and

...I- ij without taking numerical time derivatives have been

developed [74, 25, 170]. These methods recast the time derivatives of I-ij as spatial derivativesof hydrodynamic quantities computed in the collapse simulation (including the density, velocities,and gravitational potential). Thus, instantaneous values for

..I-ij and

...I- ij can be computed on a

single numerical time slice (see [170, 271] for details). Note that FHH define hpk as the maximum

value of the rms strain h =√〈h2

+ + h2×〉, where angular brackets indicate that averages have been

taken over both wavelength and viewing angle on the sky.Errors resulting from the neglect of general relativistic effects (in collapse evolutions as a whole

and in GW emission estimations like the quadrupole approximation) are of order (v/c)2. Theseerrors are typically ∼ 10% for neutron star remnants of AIC, < 30% for neutron star remnantsof massive stellar collapse, and > 30% for black hole remnants. Neglect of general relativity inrotational collapse studies is of special concern because relativistic effects counteract the stabilizingeffects of rotation (see Section 3.4).

Because the code used in the collapse simulations examined by FHH [82] was axisymmetric,their use of the numerical quadrupole approximation discussed above does not account for GWemission that may occur due to non-axisymmetric mass flow. The GWs computed directly fromtheir simulations come only from polar oscillations (which are significant when the mass flow duringcollapse [or explosion] is largely aspherical).

In order to predict the GW emission produced by non-axisymmetric instabilities, FHH employedrough analytical estimates. The expressions they used to approximate the rms strain h and thepower P = dEGW/dt of the GWs emitted by a star that has encountered the bar-mode instabilityare

hbar =

√3245

G

c4

mr2ω2

d(2)

andPbar =

3245

G

c5m2r4ω6. (3)

Here m, 2r, and ω are the mass, length, and angular frequency of the bar and d is the distance tothe source. FHH vary the mass m assumed to be enclosed by the bar (which has a correspondinglength 2r) and compute the characteristics of the GW emission as a function of this enclosed mass.For simplicity, FHH assumed that a fragmentation instability will cause a star to break into twoclumps (although more clumps could certainly be produced). Their estimates for the rms strainand power radiated by the orbiting binary fragments are

hbin =

√1285

G

c4

mr2ω2

d(4)

andPbin =

1285

G

c5m2r4ω6. (5)

Here m is the mass of a single fragment, 2r is the separation of the fragments, and ω is their orbitalfrequency.

For their computation of the GWs radiated via r-modes, FHH used the method of Ho andLai [115] (which assumes αmax = 1) and calculated only the emission from the dominant m = 2mode. This approach is detailed in FHH. If the neutron star mass and initial radius are taken tobe 1.4M and 12.53 km, respectively, the resulting formula for the average GW strain is

h(t) = 1.8× 10−24α( νs

1 kHz

) (20 Mpc

d

), (6)

where α is the mode amplitude and νs is the spin frequency.

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Gravitational Waves from Gravitational Collapse 13

FHH’s numerical quadrupole estimate of the GWs from polar oscillations in the AIC simulationof Fryer et al. [82] predicts a peak dimensionless amplitude hpk = 5.9× 10−24 (for d = 100 Mpc).The energy EGW = 3 × 1045 erg is emitted at a frequency of fGW ≈ 50 Hz. This amplitude isabout an order of magnitude too small to be observed by the advanced LIGO-II detector. Thesensitivity curve for the broadband configuration of the LIGO-II detector is shown in Figure 2(see Appendix A of [86] for details on the computation of this curve). Note that the characteristicstrain h is plotted along the vertical axis in Figure 2 (and in LISA’s sensitivity curve, shown inFigure 18). For burst sources, h = hpk. For sources that persist for N cycles, h =

√Nhpk.

Figure 2: A comparison between the GW amplitude h(f) for various sources and the LIGO-IIsensitivity curve. See the text for details regarding the computations of h. The AIC sources areassumed to be located at a distance of 100 Mpc; the SNe sources at 10 Mpc; and the Population IIIsources at a luminosity distance of ∼ 50 Gpc. Secular bar-mode sources are identified with an (s),dynamical bar-modes with a (d).

The simulation of Fryer et al. [82] considered by FHH does produce an object with an off-centerdensity maximum. However, because the maximum β reached in the AIC simulation of Fryer etal. was < 0.06 (and because the degree of differential rotation present in the remnant was low), thecollapsing object is not likely to encounter dynamical rotational instabilities. The choice of initialangular momentum J for the progenitor in an AIC simulation can affect this outcome. Fryer etal.’s choice of J = 1049 g cm2 s−1 is such that the period of the white dwarf progenitor is 10 s lessthan the shortest observed period 30 s of a cataclysmic variable white dwarf [139]. Higher angularmomenta are certainly possible. If higher values of J exist in accreting white dwarfs, bar-modeinstabilities may be more likely to occur (see the discussion of work by Liu and Lindblom below).

According to FHH, the remnant of this AIC simulation will be susceptible to r-mode growth.Assuming αmax ∼ 1 (which is likely not physical; see Section 2.3), they predict EGW > 1052 erg.FHH compute h(fGW) for coherent observation of the neutron star as it spins down over the courseof a year. For a neutron star located at a distance of 100 Mpc, this track is always below the LIGO-II noise curve. The point on this track with the maximum h, which corresponds to the beginning

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of r-mode evolution, is shown in Figure 2. The track moves down and to the left ( i.e., h and fdecrease) in this figure as the r-mode evolution continues.

2.4.2 Analyses of equilibrium representations of collapse

In addition to full hydrodynamics collapse simulations, many studies of gravitational collapse haveused hydrostatic equilibrium models to represent stars at various stages in the collapse process.Some investigators use sequences of equilibrium models to represent snapshots of the phases ofcollapse (e.g., [16, 185, 156]). Others use individual equilibrium models as initial conditions forhydrodynamical simulations (e.g., [231, 192, 184, 49]). Such simulations represent the approximateevolution of a model beginning at some intermediate phase during collapse or the evolution of acollapsed remnant. These studies do not typically follow the intricate details of the collapse itself.Instead, their goals include determining the stability of models against the development of non-axisymmetric modes and estimation of the characteristics of any resulting GW emission.

Liu and Lindblom [156, 155] have applied this equilibrium approach to AIC. Their investi-gation began with a study of equilibrium models built to represent neutron stars formed fromAIC [156]. These neutron star models were created via a two-step process, using a Newtonianversion of Hachisu’s self-consistent field method [98]. Hachisu’s method ensures that the forces dueto the centrifugal and gravitational potentials and the pressure are in balance in the equilibriumconfiguration.

Liu and Lindblom’s process of building the nascent neutron stars began with the constructionof rapidly rotating, pre-collapse white dwarf models. Their Models I and II are C-O white dwarfswith central densities ρc = 1010 and 6×109 g cm−3, respectively (recall this is the range of densitiesfor which AIC is likely for C-O white dwarfs). Their Model III is an O-Ne-Mg white dwarf that hasρc = 4×109 g cm−3 (recall this is the density at which collapse is induced by electron capture). Allthree models are uniformly rotating, with the maximum allowed angular velocities. The models’values of total angular momentum are roughly 3 – 4 times that of Fryer et al.’s AIC progenitorModel 3 [82]. The realistic equation of state used to construct the white dwarfs is a Coulombcorrected, zero temperature, degenerate gas equation of state [210, 52].

In the second step of their process, Liu and Lindblom [156] built equilibrium models of thecollapsed neutron stars themselves. The mass, total angular momentum, and specific angular mo-mentum distribution of each neutron star remnant is identical to that of its white dwarf progenitor(see Section 3 of [156] for justification of the specific angular momentum conservation assumption).These models were built with two different realistic neutron star equations of state.

Liu and Lindblom’s cold neutron star remnants had values of the stability parameter β rangingfrom 0.23 – 0.26. It is interesting to compare these results with those of Zwerger and Muller [271].Zwerger and Muller performed axisymmetric hydrodynamics simulations of stars with polytropicequations of state (P ∝ ρΓ). Their initial models were Γ = 4/3 polytropes, representative ofmassive white dwarfs. All of their models started with ρc = 1010 g cm−3. Their model thatwas closest to being in uniform rotation (A1B3) had 22% less total angular momentum thanLiu and Lindblom’s Model I. The collapse simulations of Zwerger and Muller that started withmodel A1B3 all resulted in remnants with values of β < 0.07. Comparison of the results of thesetwo studies could indicate that the equation of state may play a significant role in determiningthe structure of collapsed remnants. Or it could suggest that the assumptions employed in thesimplified investigation of Liu and Lindblom are not fully appropriate. Zwerger and Muller’s workwill be discussed in much more detail in Section 3, as it was performed in the context of corecollapse supernovae.

In a continuation of the work of Liu and Lindblom, Liu [155] used linearized hydrodynamicsto perform a stability analysis of the cold neutron star AIC remnants of Liu and Lindblom [156].He found that only the remnant of the O-Ne-Mg white dwarf (Liu and Lindblom’s Model III)

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developed the dynamical bar-mode (m = 2) instability. This model had an initial β = 0.26. Notethat the m = 1 mode, observed by others to be the dominant mode in unstable models with valuesof β much lower than 0.27 [247, 260, 192, 49], did not grow in his simulation. Because Liu andLindblom’s Models I and II had lower values of β, Liu identified the onset of instability for neutronstars formed via AIC as βd ≈ 0.25.

Liu estimated the peak amplitude of the GWs emitted by the Model III remnant to be hpk ≈1.4× 10−24 and the LIGO-II signal-to-noise ratio (for a persistent signal like that seen in the workof [184] and [34]) to be S/N ≤ 3 (for fGW ≈ 450 Hz). These values are for a source located at100 Mpc. He also predicted that the timescale for gravitational radiation to carry away enoughangular momentum to eliminate the bar-mode is τGW ∼ 7 s (∼ 3×103 cycles). Thus, h ∼ 8×10−23.(Note that this value for h is merely an upper limit as it assumes that the amplitude and frequencyof the GWs do not change over the 7 s during which they are emitted. Of course, they will changeas angular momentum is carried away from the object via GW emission.) Such a signal may bemarginally detectable with LIGO-II (see Figure 2). Details of the approximations on which theseestimates are based can be found in [155].

Liu cautions that his results hold if the magnetic field of the proto-neutron star is B ≤ 1012 G. Ifthe magnetic field is larger, then it may have time to suppress some of the neutron star’s differentialrotation before it cools. This would make bar formation less likely. Such a large field could onlyresult if the white dwarf progenitor’s B field was ≥ 108 G. Observation-based estimates suggestthat about 25% of white dwarfs in interacting close binaries (cataclysmic variables) are magneticand that the field strengths for these stars are ∼ 107 – 3× 108 G [256].

2.5 Going further

The AIC scenario is generally discussed in terms of the collapse of an accreting white dwarf to aneutron star. However, Shibata, Baumgarte, and Shapiro have examined the collapse of a rotating,supramassive neutron star to a black hole [224]. Such supramassive neutron stars (with massesgreater than the maximum mass for a nonrotating neutron star) could be formed and pushed tocollapse via accretion from a binary companion. They performed 3D, fully general relativistichydrodynamics simulations of uniformly rotating neutron stars. Dynamical non-axisymmetricinstabilities (such as the bar-mode) did not have time to grow in their simulations prior to blackhole formation. Differentially rotating neutron star progenitors could have higher values of β thanthe uniformly rotating models used in this study and may be susceptible to non-axisymmetricinstabilities on a shorter timescale.

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3 Collapse of Massive Stars

3.1 Collapse scenario

Stars with mass greater than ∼ 8 – 10 M will undergo core collapse at the end of their thermonu-clear burning life cycles. The gravitational energy released during the collapse is believed to bethe power source behind a large subset of supernovae (for reviews, see [8, 23, 79, 81, 143]). Corecollapse SNe include Types II and Ib/Ic. SNe Ib/Ic are distinguished from Type II SNe by the ab-sence of hydrogen in their spectra. SNe Ib/Ic are thought to result from the collapse of the cores ofmassive stars that have lost their hydrogen envelopes (and possibly part of their helium envelopes)by stellar winds or by mass transfer. The SN Ib/Ic progenitors that lose their outer envelopes viastellar winds are known as Wolf–Rayet stars and have initial masses & 30 M [108, 266]; those thatundergo mass loss via mass transfer in binaries can have progenitor masses as low as 12 M [194].

If the stellar mass is less than ∼ 20 – 25 M [78, 89, 86, 108], it is believed that the starwill produce a strong supernova, leaving behind a neutron star remnant. This section studiesthe gravitational waves produced during the stellar collapse and supernova explosion mechanism.Without mass loss, stars above 40 – 50 M are thought to collapse directly to black holes withoutproducing a supernova explosion [78, 108]. If rotating, the collapsing stars may form an accretiondisk around their black hole core. One of the leading models for long-duration gamma-ray bursts(the “collapsar” engine) argues that the energy extracted from the disk or the black hole spincan drive a relativistic jet [262]. We will discuss these collapsars and their resultant gravitationalwaves further in Section 4. However, if we include the effects of winds, these stars lose much oftheir mass through winds during their nuclear-burning lifetimes. Their fate will be closer to thatof a 20 M and are likely to produce supernovae and, at least initially, neutron stars, not blackholes [89, 108]. If the progenitor’s mass is in the range 20 − 25 . M . 40 – 50 M, the entirestar is not ejected in the SN explosion. More than 2M will fall back onto the nascent neutronstar and lead to black hole formation. These objects also produce a variant of the collapsar enginefor gamma-ray bursts (GRBs). Although the discussion of GW production from the collapse andsupernova explosion phase will be discussed in this section, the GWs produced during the fallbackand black hole accretion disk phase will be discussed in Section 4. Note that the limits on theprogenitor masses quoted in this paragraph (especially the 40 – 50 M lower limit for direct blackhole formation) are uncertain because the progenitor mass dependence of the neutrino explosionmechanism (see below) is unknown [103, 178].

The massive iron cores of SN II/Ib/Ic progenitors are supported by both thermal and electrondegeneracy pressures. The density and temperature of such a core will eventually rise, due to thebuild up of matter consumed by thermonuclear burning, to the point where electron capture andphotodissociation of nuclei begin. Dissociation lowers the photon and electron temperatures andthereby reduces the core’s thermal support [71]. Electron capture reduces the electron degeneracypressure. One or both of these processes will trigger the collapse of the core. The relative impor-tance of dissociation and electron capture in instigating collapse is determined by the mass of thestar [71]. The more massive the core, the bigger is the role played by dissociation.

Approximately 70% of the inner portion of the core collapses homologously and subsonically.The outer core collapses at supersonic speeds [71, 170]. The maximum velocity of the outer regionsof the core reaches ∼ 7×104 km s−1. It takes just 1 s for an earth-sized core to collapse to a radiusof 50 km [8].

The inward collapse of the core is halted by nuclear forces when its central density ρc is 2 – 10times the density of nuclear material [13, 12]. The core overshoots its equilibrium position andbounces. A shock wave is formed when the supersonically infalling outer layers hit the reboundinginner core. If the inner core pushes the shock outwards with energy E > 1051 erg (supplied bythe binding energy of the nascent neutron star), then the remainder of the star can be ejected

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in about 20 ms [8]. This so-called “prompt explosion” mechanism has been succesful numericallyonly when a very soft supra-nuclear equation of state is used in conjunction with a relatively smallcore (M . 1.35M, derived from a very low mass progenitor) and a large portion of the collapseproceeds homologously [23, 173]. Inclusion of general relativistic effects in collapse simulations canincrease the success of the prompt mechanism in some cases [13, 237].

Both dissociation of nuclei and electron capture can reduce the ejection energy, causing theprompt mechanism to fail. The shock will then stall at a radius in the range 100 – 200 km. Colgateand White [55] suggested that energy from neutrinos emitted by the collapsed core could revive thestalled shock. (See Burrows and Thompson [44] for a review of core collapse neutrino processes.)However, their simulations did not include the physics necessary to accurately model this “delayedexplosion” mechanism. Wilson and collaborators were the first to perform collapse simulations withsuccessful delayed ejections [30, 29, 257, 24, 259, 258]. However, their 1-dimensional simulations andthose of others had difficulty producing energies high enough to match observations [53, 36, 131].It was not until Herant and collaborators modelled the collapse and bounce phase with neutrino-transport in 2-dimensions that convection began to be accepted as a necessary puzzle piece inunderstanding the supernova explosion mechanism [112].

Observations of SN 1987a show that significant mixing occurred during this supernovae (seeArnett et al. [8] for a review). Such mixing can be attributed to nonradial motion resultingfrom fluid instabilities. Convective instabilities play a significant role in the current picture ofthe delayed explosion mechanism. The outer regions of the nascent neutron star are convectivelyunstable after the shock stalls (for an interval of 10 – 100 ms after bounce) due to the presenceof negative lepton and energy gradients [173]. This has been confirmed by both 2D and 3Dsimulations [112, 42, 132, 173, 167, 78, 83, 128, 130, 197, 91, 85, 26, 39, 92, 80, 212, 253]. Convectivemotion is more effective at transporting neutrinos out of the proto-neutron star than is diffusion.Less than 10% of the neutrinos emitted by the neutron star need to be absorbed and convertedto kinetic energy for the shock to be revived [173]. The “hot bubble” region above the surfaceof the neutron star also has been shown to be convectively unstable [53, 23, 54, 91]. Janka andMuller have demonstrated that convection in this region only aids the explosion if the neutrinoluminosity is in a narrow region [133]. Some simulations that include advanced neutrino transportmethods have cast doubt on the ability of convection to ensure the success of the delayed explosionmechanism [168, 130, 197, 39] and this problem is far from solved. But progress not only in neutrinotransport, but in understanding new features in the convection [26, 43, 75] some including the effectof magnetic fields is leading to new ideas about the supernova mechanism [3]. We will discuss theeffects of this new physics on the GW signal at the end of this section.

In addition to the mixing seen in SN 1987a, observations of (i) polarization in the spectra ofseveral core collapse SNe, (ii) jets in the Cas A remnant, and (iii) kicks in neutron stars suggestthat supernovae are inherently aspherical (see [8, 3, 116, 122, 121] and references therein). Notethat these asphericities could originate in the central explosion mechanism itself and/or the mecha-nism(s) for energy transfer between the core and ejecta [130]. If due to the mechanism itself, theseasymmetries may provide clues into the true engine behind supernova explosions. Already, theobservations partly motivated the multi-dimensional studies of convection in the delayed explosionmechanism as well as work on magnetic field engines [17, 255, 3]. Observations have also driventhe work on jets and collapsars 4. Hoflich et al. [116] have argued that low velocity jets stalledinside SN envelopes can account for the observed asymmetries. Hungerford and collaborators haveargued that the asymmetries required are not so extreme [122, 121], arguments that have now beenconfirmed [138]. This debate is crucial to our understanding of the supernova mechanism. If jetsare required, magnetic field mechanisms are the likely source of the asymmetry. If jets are notrequired, the convective engine can produce asymmetries via a number of channels from low modeconvection [26, 212, 43] to rotation(e.g.,[83]) to asymmetric collapse [41, 80]. Most GW emissioncalculations of stellar collapse have focused on the collapse of rotating, massive stars with rotation

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periods at least as high as those assumed by Fryer & Heger [83], implicitly assuming that theasymmetries are driving by rotation.

3.2 Formation rate

Type II/Ib/Ic supernovae are observed to occur in only spiral and irregular galaxies. The mostthorough computation of SN rates is that of Cappellaro et al. [46]. Their sample includes 137 SNefrom five different SN searches. They determined that the core collapse SN rate in the Galaxy is6 × 10−3 – 1.6 × 10−2 yr−1. Thus, a reasonable occurrence rate can be found for an observationdistance of 10 Mpc. (Note that a infrared survey estimates that the rate may actually be an orderof magnitude higher [161].) What this rate might exclude are those stars that collapse to formneutron stars, but have sufficiently weak explosions that they eject very little nickel and hence arenot very bright. Including these systems as well will increase the total number of stellar collapsesinitially into neutron stars by less than 5 – 40% [89]. However, it does include those stars thatcollapse to black holes directly and form GRBs (producing supernova-like outbursts). We willdiscuss the GWs from such collapses in Section 4. Excluding them from this sample will notchange the above numbers.

3.3 GW emission mechanisms

Gravitational radiation will be emitted during the collapse/explosion of a core collapse SN dueto the star’s changing quadrupole moment. A rough description of the possible evolution of thequadrupole moment is given in the remainder of this paragraph. During the first 100 – 250 ms ofthe collapse, as the core contracts and flattens, the magnitude of the quadrupole moment I-ij willincrease. The contraction speeds up over the next 20 ms and the density distribution becomes acentrally condensed torus [170]. In this phase the core’s shrinking size dominates its increasingdeformation and the magnitude of I-ij decreases. As the core bounces, I-ij changes rapidly dueto the deceleration and rebound. If the bounce occurs because of nuclear pressure, its timescalewill be < 1 ms. If centrifugal forces play a role in halting the collapse, the bounce can last up toseveral ms [170]. The magnitude of I-ij will increase due to the core’s expansion after bounce. Asthe resulting shock moves outwards, the unshocked portion of the core will undergo oscillations,causing I-ij to oscillate as well. The shape of the core, the depth of the bounce, the bouncetimescale, and the rotational energy of the core all strongly affect the GW emission. For furtherdetails see [71, 170].

Convectively driven inhomogeneities in the density distribution of the outer regions of thenascent neutron star and anisotropic neutrino emission are other sources of GW emission duringthe collapse/explosion (see [41, 176, 80, 158] for reviews).

As discussed in the case of AIC, global rotational instabilities (such as the m = 2 bar-mode)may develop during the collapse itself or in a neutron star remnant. A neutron star remnantwill likely also be susceptible to the radiation reaction driven r-modes. Both of these types ofinstabilities will emit GWs, as will a fragmentation instability if one occurs. See Section 2.3 forfurther details regarding these instabilities.

3.4 Numerical predictions of GW emission

3.4.1 Historical investigations

The collapse of the progenitors of core collapse supernovae has been investigated as a source ofgravitational radiation for more than three decades. In an early study published in 1971, Ruffiniand Wheeler [203] identified mechanisms related to core collapse that could produce GWs andprovided order-of-magnitude estimates of the characteristics of such emission.

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Quantitative computations of GW emission during the infall phase of collapse were performedby Thuan and Ostriker [245] and Epstein and Wagoner [66, 65], who simulated the collapse ofoblate dust spheroids. Thuan and Ostriker used Newtonian gravity and computed the emittedradiation in the quadrupole approximation. Epstein and Wagoner discovered that post-Newtonianeffects prolonged the collapse and thus lowered the GW luminosity. Subsequently, Novikov [188]and Shapiro and Saenz [218, 204] included internal pressure in their collapse simulations and werethus able to examine the GWs emitted as collapsing cores bounced at nuclear densities. Thequadrupole GWs from the ringdown of the collapse remnant were initially investigated by theperturbation study of Turner and Wagoner [249] and later by Saenz and Shapiro [205, 206].

Muller [171] calculated the quadrupole GW emission from 2D axisymmetric collapse basedon the Newtonian simulations of Muller and Hillebrandt [174] (these simulations used a realisticequation of state and included differential rotation). He found that differential rotation enhancedthe efficiency of the GW emission.

Stark and Piran [233, 193] were the first to compute the GW emission from fully relativisticcollapse simulations, using the ground-breaking formalism of Bardeen and Piran [11]. They followedthe (pressure-cut induced) collapse of rotating polytropes in 2D. Their work focused in part onthe conditions for black hole formation and the nature of the resulting ringdown waveform, whichthey found could be described by the quasi-normal modes of a rotating black hole. In each of theirsimulations, less than 1% of the gravitational mass was converted to GW energy.

Seidel and collaborators also studied the effects of general relativity on the GW emission duringcollapse and bounce [215, 216]. They employed a perturbative approach, valid only in the slowlyrotating regime.

The gravitational radiation from non-axisymmetric collapse was investigated by Detweiler andLindblom, who used a sequence of non-axisymmetric ellipsoids to represent the collapse evolu-tion [57]. They found that the radiation from their analysis of non-axisymmetric collapse wasemitted over a more narrow range of frequency than in previous studies of axisymmetric collapse.

For further discussion of the first two decades of study of the GW emission from stellar collapsesee [72]. In the remainder of Section 3.4, more recent investigations will be discussed.

3.4.2 Axisymmetric simulations

The core collapse simulations of Monchmeyer et al. began with better iron core models and a morerealistic microphysical treatment (including a realistic equation of state, electron capture, and asimple neutrino transport scheme) than any previous study of GW emission from axisymmetricstellar core collapse [170]. The shortcomings of their investigation included initial models thatwere not in rotational equilibrium, an equation of state that was somewhat stiff in the subnuclearregime, and the use of Newtonian gravity. Each of their four models had a different initial angularmomentum profile. The rotational energies of the models ranged from 0.1 – 0.45 of the maximumpossible rotational energy.

The collapses of three of the four models of Monchmeyer et al. were halted by centrifugalforces at subnuclear densities. This type of low ρc bounce had been predicted by Shapiro andLightman [219] and Tohline [246] (in the context of the “fizzler” scenario for failed supernovae; seealso [105, 106, 124]), and had been observed in earlier collapse simulations [175, 238]. Monchmeyerand collaborators found that a bounce caused by centrifugal forces would last for several ms,whereas a bounce at nuclear densities would occur in < 1 ms. They also determined that a subnu-clear bounce produced larger amplitude oscillations in density and radius, with larger oscillationperiods, than a bounce initiated by nuclear forces alone. They pointed out that these differencesin timescale and oscillatory behavior should affect the GW signal. Therefore, the GW emissioncould indicate whether the bounce was a result of centrifugal or nuclear forces.

Monchmeyer et al. identified two different types of waveforms in their models (computed using

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the numerical quadrupole approximation discussed in Section 2.4). The waveforms they categorizedas Type I (similar to those observed in previous collapse simulations [171, 74]) are distinguishedby a large amplitude peak at bounce and subsequent damped ringdown oscillations. They notedthat Type I signals were produced by cores that bounced at nuclear densities (or bounced atsubnuclear densities if the cores had small ratios of radial kinetic to rotational kinetic energies).The quadrupole gravitational wave amplitude AE2

20 for a Type I waveform is shown in Figure 3(see [241, 271] for expressions relating AE2

20 to h). The waveforms identified as Type II exhibitseveral maxima, which result from multiple bounces (see Figure 4 for an example of a Type IIwaveform). Note that the waveforms displayed in Figures 3 and 4 are from the study of Zwergerand Muller [271], discussed below.

Figure 3: Type I waveform (quadrupole amplitude AE220 as a function of time) from one of Zwerger

and Muller’s [271] simulations of a collapsing polytrope. The vertical dotted line marks the timeat which the first bounce occurred. (Figure 5d of [271]; used with permission.)

The model of Monchmeyer et al. that bounced due to nuclear forces had the highest GWamplitude of all of their models, hpk ∼ 10−23 for a source distance d = 10 Mpc, and the largestemitted energy EGW ∼ 1047 erg. The accompanying power spectrum peaked in the frequencyrange 5× 102 – 103 Hz.

The most extensive Newtonian survey of the parameter space of axisymmetric, rotational corecollapse is that of Zwerger and Muller [271]. They simulated the collapse of 78 initial modelswith varying amounts of rotational kinetic energy (reflected in the initial value of the stabilityparameter βi), differential rotation, and equation of state stiffness. In order to make this largesurvey tractable, they used a simplified equation of state and did not explicitly account for electroncapture or neutrino transport. Their initial models were constructed in rotational equilibrium viathe method of Eriguchi and Muller [67]. The models had a polytropic equation of state, withinitial adiabatic index Γi = 4/3. Collapse was induced by reducing the adiabatic index to a value

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Figure 4: Type II waveform (quadrupole amplitude AE220 as a function of time) from one of Zwerger

and Muller’s [271] simulations of a collapsing polytrope. The vertical dotted line marks the timeat which bounce occurred. (Figure 5a of [271]; used with permission.)

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Γr in the range 1.28 – 1.325. The equation of state used during the collapse evolution had bothpolytropic and thermal contributions (note that simulations using more sophisticated equations ofstate get similar results [144]).

Figure 5: Type III waveform (quadrupole amplitude AE220 as a function of time) from one of Zwerger

and Muller’s [271] simulations of a collapsing polytrope. The vertical dotted line marks the timeat which bounce occurred. (Figure 5e of [271]; used with permission).

The major result of Zwerger and Muller’s investigation was that the signal type of the emittedgravitational waveform in their runs was determined by the stiffness of the equation of stateof the collapsing core (i.e., the value of Γr). In their simulations, Type I signals (as labelled byMonchmeyer et al. [170]) were produced by models with relatively soft equations of state, Γr . 1.31.Type II signals were produced by the models with stiffer equations of state, Γr & 1.32. They founda smooth transition between these signal types if Γr was increased while all other parameterswere held fixed. They also observed another class of signal, Type III, for their models with thelowest Γr (= 1.28). Type III waveforms have a large positive peak just prior to bounce, a smallernegative peak just after bounce, and smaller subsequent oscillations with very short periods (seeFigure 5). Type III signals were not observed in the evolution of strongly differentially rotatingΓr = 1.28 models and were also not seen in subsequent investigations [144, 190]. Their waveformswere computed with the same technique used in [170].

In contrast to the results of Monchmeyer et al. [170], in Zwerger and Muller’s investigation thevalue of ρc at bounce did not determine the signal type. Instead, the only effect on the waveformdue to ρc was a decrease in hpk in models that bounced at subnuclear densities. The effect ofthe initial value of βi on hpk was non-monotonic. For models with βi . 0.1, hpk increased withincreasing βi. This is because the deformation of the core is larger for faster rotators. However,for models with larger βi, hpk decreases as βi increases. These models bounce at subnuclear

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densities. Thus, the resulting acceleration at bounce and the GW amplitude are smaller. Zwergerand Muller found that the maximum value of hpk for a given sequence was reached when ρc atbounce was just less than ρnuc. The degree of differential rotation did not have a large effect onthe emitted waveforms computed by Zwerger and Muller. However, they did find that models withsoft equations of state emitted stronger signals as the degree of differential rotation increased.

The models of Zwerger and Muller that produced the largest GW signals fell into two categories:those with stiff equations of state and βi < 0.01; and those with soft equations of state, βi ≥ 0.018,and large degrees of differential rotation. The GW amplitudes emitted during their simulationswere in the range 4 × 10−25 . h . 4 × 10−23, for d = 10 Mpc (the model with the highest h isidentified in Figure 2). The corresponding energies ranged from 1044 . EGW . 1047 erg. Thepeaks of their power spectra were between 500 Hz and 1 kHz. Such signals would fall just outsideof the range of LIGO-II. Magnetic fields lower these amplitudes by ∼ 10% [145], but realistic stellarprofiles can lower the amplitudes by a factor of & 4− 10 [190, 177], restricting the detectability ofsupernovae to within our Galaxy (. 10 kpc).

Yamada and Sato [264] used techniques very similar to those of Zwerger and Muller [271] intheir core collapse study. Their investigation revealed that the hpk for Type I signals becamesaturated when the dimensionless angular momentum of the collapsing core, q = J/(2GM/c),reached ∼ 0.5. They also found that hpk was sensitive to the stiffness of the equation of state fordensities just below ρnuc. The characteristics of the GW emission from their models were similarto those of Zwerger and Muller.

3.4.3 Non-axisymmetric simulations

The GW emission from non-axisymmetric hydrodynamics simulations of stellar collapse was firststudied by Bonazzola and Marck [163, 28]. They used a Newtonian, pseudo-spectral hydrodynamicscode to follow the collapse of polytropic models. Their simulations covered only the pre-bouncephase of the collapse. They found that the magnitudes of hpk in their 3D simulations were within afactor of two of those from equivalent 2D simulations and that the gravitational radiation efficiencydid not depend on the equation of state.

The first 3D hydrodynamics collapse simulations to study the GW emission well beyond thecore bounce phase were performed by Rampp, Muller, and Ruffert [198]. These authors startedtheir Newtonian simulations with the only model (A4B5G5) of Zwerger and Muller [271] that hada post-bounce value for the stability parameter β = 0.35 that significantly exceeded 0.27 (recallthis is the value at which the dynamical bar instability sets in for MacLaurin spheroid-like models).This model had the softest EOS (Γr = 1.28), highest βi = 0.04, and largest degree of differentialrotation of all of Zwerger and Muller’s models. The model’s initial density distribution had anoff-center density maximum (and therefore a torus-like structure). Rampp, Muller, and Ruffertevolved this model with a 2D hydrodynamics code until its β reached ≈ 0.1. At that point, 2.5 msprior to bounce, the configuration was mapped onto a 3D nested cubical grid structure and evolvedwith a 3D hydrodynamics code.

Before the 3D simulations started, non-axisymmetric density perturbations were imposed toseed the growth of any non-axisymmetric modes to which the configuration was unstable. When theimposed perturbation was random (5% in magnitude), the dominant mode that arose was m = 4.The growth of this particular mode was instigated by the cubical nature of the computationalgrid. When an m = 3 perturbation was imposed (10% in magnitude), three clumps developedduring the post-bounce evolution and produced three spiral arms. These arms carried mass andangular momentum away from the center of the core. The arms eventually merged into a bar-likestructure (evidence of the presence of the m = 2 mode). Significant non-axisymmetric structurewas visible only within the inner 40 km of the core. Their simulations were carried out to ∼ 14 msafter bounce.

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24 Chris L. Fryer and Kimberly C. B. New

The amplitudes of the emitted gravitational radiation (computed in the quadrupole approxi-mation) were only ∼ 2% different from those observed in the 2D simulation of Zwerger and Muller.Because of low angular resolution in the 3D runs, the energy emitted was only 65% of that emittedin the corresponding 2D simulation.

The findings of Centrella et al. [49] indicate it is possible that some of the post-bounce config-urations of Zwerger and Muller, which have lower values of β than the model studied by Rampp,Muller, and Ruffert [198], may also be susceptible to non-axisymmetric instabilities. Centrella etal. have performed 3D hydrodynamics simulations of Γ = 1.3 polytropes to test the stability ofconfigurations with off-center density maxima (as are present in many of the models of Zwergerand Muller [271]). The simulations carried out by Centrella and collaborators were not full collapsesimulations, but rather began with differentially rotating equilibrium models. These simulationstracked the growth of any unstable non-axisymmetric modes that arose from the initial 1% ran-dom density perturbations that were imposed. Their results indicate that such models can becomedynamically unstable at values of β & 0.14. The observed instability had a dominant m = 1mode. Centrella et al. estimate that if a stellar core of mass M ∼ 1.4M and radius R ∼ 200 kmencountered this instability, the values of hpk from their models would be ∼ 2× 10−24 – 2× 10−23,for d = 10 Mpc. The frequency at which hpk occurred in their simulations was ∼ 200 Hz. Thisinstability would have to persist for at least ∼ 15 cycles to be detected with LIGO-II.

Brown [35] carried out an investigation of the growth of non-axisymmetric modes in post-bounce cores that was similar in many respects to that of Rampp, Muller, and Ruffert [198]. Heperformed 3D hydrodynamical simulations of the post-bounce configurations resulting from 2Dsimulations of core collapse. His pre-collapse initial models are Γ = 4/3 polytropes in rotationalequilibrium. The differential rotation laws used to construct Brown’s initial models were motivatedby the stellar evolution study of Heger, Langer, and Woosley [109]. The angular velocity profilesof their pre-collapse progenitors were broad and Gaussian-like. Brown’s initial models had peakangular velocities ranging from 0.8 – 2.4 times those of [109]. The model evolved by Rampp, Muller,and Ruffert [198] had much stronger differential rotation than any of Brown’s models. To inducecollapse, Brown reduced the adiabatic index of his models to Γ = 1.28, the same value used by [198].

Brown found that β increased by a factor . 2 during his 2D collapse simulations. This is muchless than the factor of ∼ 9 observed in the model studied by Rampp, Muller, and Ruffert [198].This is likely a result of the larger degree of differential rotation in the model of Rampp et al.

Brown performed 3D simulations of the two most rapidly rotating of his post-bounce models(models Ω24 and Ω20, both of which had β > 0.27 after bounce) and of the model of Rampp et al.(which, although it starts out with β = 0.35, has a sustained β < 0.2). Brown refers to the Ramppet al. model as model RMR. Because Brown’s models do not have off-center density maxima, theyare not expected to be unstable to the m = 1 mode observed by Centrella et al. [49]. He imposedrandom 1% density perturbations at the start of all three of these 3D simulations (note that thisperturbation was of a much smaller amplitude than those imposed by [198]).

Brown’s simulations determined that both his most rapidly rotating model Ω24 (with post-bounce β > 0.35) and model RMR are unstable to growth of the m = 2 bar-mode. However, hismodel Ω20 (with post-bounce β > 0.3) was stable. Brown observed no dominant m = 3 or m = 4modes growing in model RMR at the times at which they were seen in the simulations of Rampp etal. This suggests that the mode growth in their simulations was a result of the large perturbationsthey imposed. The m = 2 mode begins to grow in model RMR at about the same time as Ramppet al. stopped their evolutions. No substantial m = 1 growth was observed.

The results of Brown’s study indicate that the overall β of the post-bounce core may not be agood diagnostic for the onset of instability. He found, as did Rampp, Muller, and Ruffert [198],that only the innermost portion of the core (with ρ > 1010 g cm−3) is susceptible to the bar-mode.This is evident in the stability of his model Ω20. This model had an overall β > 0.3, but an innercore with βic = 0.15. Brown also observed that the β of the inner core does not have to exceed 0.27

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for the model to encounter the bar-mode. Models Ω24 and RMR had βic ≈ 0.19. He speculatesthat the inner cores of these later two models may be bar-unstable because interaction with theirouter envelopes feeds the instability or because βd < 0.27 for such configurations.

Fryer and Warren [91] performed the first 3D collapse simulations to follow the entire collapsethrough explosion. They used a smoothed particle hydrodynamics code, a realistic equation ofstate, the flux-limited diffusion approximation for neutrino transport, and Newtonian sphericalgravity. Their initial model was nonrotating. Thus, no bar-mode instabilities could develop duringtheir simulations. The only GW emitting mechanism present in their models was convection inthe core. The maximum amplitude h of this emission, computed in the quadrupole approximation,was ∼ 3× 10−26, for d = 10 Mpc [88]. In later work, Fryer & Warren [92] included full Newtoniangravity through a tree algorithm and studied the rotating progenitors from Fryer & Heger [83].By the launch of the explosion, no bar instabilities had developed. This was because of severaleffects: they used slowly rotating, but presumably realistic, progenitors [83], the explosion occuredquickly for their models (. 100 ms) and, finally, because much of the high angular momentummaterial did not make it into the inner core. These models have been further studied for the GWsignals [87]. The fastest rotating models achieved a signal of h ∼ 2 × 10−24 for d = 10 Mpc andcharacteristic frequencies of fGW ∼ 1000 Hz. For supernovae occuring within the Galaxy, such asignal is detectable by LIGO-II.

Fryer and collaborators have also modeled asymmetric collapse and asymmetric explosion cal-culations in 3 dimensions [80, 90]. These calculations will be discussed in Section 3.4.5.

The GW emission from nonradial quasinormal mode oscillations in proto-neutron stars has beenexamined by Ferrari, Miniutti, and Pons [70]. They found that the frequencies of emission fGW

during the first second after formation (600 – 1100 Hz for the first fundamental and gravity modes)are significantly lower than the corresponding frequencies for cold neutron stars and thus residein the bandwidths of terrestrial interferometers. However, for first generation interferometers todetect the GW emission from an oscillating proto-neutron star located at 10 Mpc, with a signal-to-noise ratio of 5, EGW must be ∼ 10−3 – 10−2 Mc2. It is unlikely that this much energy is storedin these modes (the collapse itself may only emit ∼ 10−7 Mc2 in gravitational waves [60]).

3.4.4 General relativistic simulations

General relativistic effects oppose the stabilizing influence of rotation in pre-collapse cores. Thus,stars that might be prevented from collapsing due to rotational support in the Newtonian limitmay collapse when general relativistic effects are considered. Furthermore, general relativity willcause rotating stars undergoing collapse to bounce at higher densities than in the Newtoniancase [239, 271, 198, 37].

The full collapse simulations of Fryer and Heger [83] are the most sophisticated axisymmetricsimulations from which the resultant GW emission has been studied [86, 88]. Fryer and Hegerinclude the effects of general relativity, but assume (for the purposes of their gravity treatment only)that the mass distribution is spherical. The GW emission from these simulations was evaluatedwith either the quadrupole approximation or simpler estimates (see below).

The work of Fryer and Heger [83] is an improvement over past collapse investigations because itstarts with rotating progenitors evolved to collapse with a stellar evolution code (which incorporatesangular momentum transport via an approximate diffusion scheme) [107], incorporates realisticequations of state and neutrino transport, and follows the collapse to late times. The values oftotal angular momentum of the inner cores of Fryer and Heger (0.95 – 1.9 × 1049 g cm2 s−1) arelower than has often been assumed in studies of the GW emission from core collapse. Note thatthe total specific angular momentum of these core models may be lower by about a factor of 10 ifmagnetic fields were included in the evolution of the progenitors [3, 232, 110].

FHH’s [86] numerical quadrupole estimate of the GWs from polar oscillations in the collapse

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simulations of Fryer and Heger [83] predicts a peak dimensionless amplitude hpk = 4.1 × 10−23

(for d = 10 Mpc), emitted at fGW ≈ 20 Hz. The radiated energy EGW ∼ 2× 1044 erg. This signalwould be just out of the detectability range of the LIGO-II detector.

The cores in the simulations of Fryer and Heger [83] are not compact enough (or rotating rapidlyenough) to develop bar instabilities during the collapse and initial bounce phases. However, theexplosion phase ejects a good deal of low angular momentum material along the poles in theirevolutions. Therefore, about 1 s after the collapse, β becomes high enough in their models toexceed the secular bar instability limit. The β of their model with the least angular momentumactually exceeds the dynamical bar instability limit as well (it contracts to a smaller radius and thushas a higher spin rate than the model with higher angular momentum). FHH (and [88]) computean upper limit (via Equation (2)) to the emitted amplitude from their dynamically unstable modelof h ∼ 3 × 10−22 (if coherent emission from a bar located at 10 Mpc persists for 100 cycles).The corresponding frequency and maximum power are fGW ≈ 103 Hz and PGW = 1053 erg s−1.LIGO-II should be able to detect such a signal (see Figure 2, where FHH’s upper limit to h forthis dynamical bar-mode is identified).

As mentioned above, the proto-neutron stars of Fryer and Heger are likely to be unstable tothe development of secular bar instabilities. The GW emission from proto-neutron stars that aresecularly unstable to the bar-mode has been examined by Lai and Shapiro [148, 146]. Becausethe timescale for secular evolution is so long, 3D hydrodynamics simulations of the nonlinear de-velopment of a secular bar can be impractical. To bypass this difficulty, Lai [146] considers onlyincompressible fluids, for which there are exact solutions for (Dedekind and Jacobi-like) bar devel-opment. He predicts that such a bar located at 10 Mpc would emit GWs with a peak characteristicamplitude h ∼ 10−21, if the bar persists for 102 – 104 cycles. The maximum fGW of the emittedradiation is in the range 102 – 103 Hz. This type of signal should be easily detected by LIGO-I(although detection may require a technique like the fast chirp transform method of Jenet andPrince [134] due to the complicated phase evolution of the emission).

Alternatively, Ou et al. [191] bypassed the long secular timescale by increasing the driving forceof the instability. They found that a bar instability was maintained for several orbits before sheerflows, producing GW emission that would have a signal-to-noise ratio greater than 8 for LIGO-IIout to 32 Mpc. A movie of this simulation is shown in Figure 6.

Figure 6: Still from a Movie showing the evolution of a secular bar instability, see Ou et al. [191]for details. (To watch the movie, please go to the online version of this review article at http:// www. livingreviews. org/ lrr-2003-2 .)

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FHH predict that a fragmentation instability is unlikely to develop during core collapse SNebecause the cores have central density maxima (see also [88]). However, they do give estimates[calculated via Equations (4, 5)] for the amplitude, power, and frequency of the emission from suchan instability: hpk ∼ 2× 10−22, PGW = 1054 erg s−1, fGW ≈ 2× 103 Hz. Again, this signal wouldfall just beyond the upper limit of LIGO-II’s frequency range.

The GW emission from r-mode unstable neutron star remnants of core collapse SNe wouldbe easily detectable if αmax ∼ 1 (which is likely not physical; see Section 2.3). Multiple GWbursts will occur as material falls back onto the neutron star and results in repeat episodes ofr-mode growth (note that a single r-mode episode can have multiple amplitude peaks [153]). FHHcalculate that the characteristic amplitude of the GW emission from this r-mode evolution tracksfrom 6 – 1×10−22, over a frequency range of 103 – 102 Hz (see Section 2.4 for details). They estimatethe emitted energy to exceed 1052 erg.

General relativity has been more fully accounted for in the core collapse studies of Dimmelmeier,Font, and Muller [58, 59, 60] and Shibata and collaborators [230], which build on the Newtonian,axisymmetric collapse simulations of Zwerger and Muller [271]. In all, they have followed the col-lapse evolution of 26 different models, with both Newtonian and general relativistic simulations.As in the work of Zwerger and Muller, the different models are characterized by varying degreesof differential rotation, initial rotation rates, and adiabatic indices. They use the conformally flatmetric to approximate the space time geometry [56] in their relativistic hydrodynamics simula-tions. This approximation gives the exact solution to Einstein’s equations in the case of sphericalsymmetry. Thus, as long as the collapse is not significantly aspherical, the approximation is rel-atively accurate. However, the conformally flat condition does eliminate GW emission from thespacetime. Because of this, Dimmelmeier, Font, and Muller used the quadrupole approximationto compute the characteristics of the emitted GW signal (see [271] for details).

The general relativistic simulations of Dimmelmeier et al. showed the three different typesof collapse evolution (and corresponding gravitational radiation signal) seen in the Newtoniansimulations of Zwerger and Muller (regular collapse – Type I signal; multiple bounce collapse –Type II signal; and rapid collapse – Type III signal). However, relativistic effects sometimes led toa different collapse type than in the Newtonian case. This is because general relativity did indeedcounteract the stabilizing effects of rotation and led to much higher bounce densities (up to 700%higher). They found that multiple bounce collapse is much rarer in general relativistic simulations(occurring in only two of their models). When multiple bounce does occur, relativistic effectsshorten the time interval between bounces by up to a factor of four. Movies of the simulations offour models from Dimmelmeier et al. [60] are shown in Figures 7, 8, 9, and 10. The four evolutionsshown include a regular collapse (Movie 7), a rapid collapse (Movie 8), a multiple bounce collapse(Movie 9), and a very rapidly and differentially rotating collapse (Movie 10). The left frames ofeach movie contain the 2D evolution of the logarithmic density. The upper and lower right framesdisplay the evolutions of the gravitational wave amplitude and the maximum density, respectively.These movies can also be viewed at [165].

Dimmelmeier et al. found that models for which the collapse type was the same in both New-tonian and relativistic simulations had lower GW amplitudes hpk in the relativistic case. This isbecause the Newtonian models were less compact at bounce and thus had material with higherdensities and velocities at larger radii. Both higher and lower values of hpk were observed in mod-els for which the collapse type changed. Overall, the range of hpk (4 × 10−24 – 3 × 10−23, for asource located at 10 Mpc) seen in the relativistic simulations was quite close to the correspondingNewtonian range. The average EGW was somewhat higher in the relativistic case (1.5 × 1047 ergcompared to the Newtonian value of 6.4×1046 erg). The overall range of GW frequencies observedin their relativistic simulations (60 – 1000 Hz) was close to the Newtonian range. They did notethat relativistic effects always caused the characteristic frequency of emission, fGW, to increase(up to five-fold). Studies of non-linear pulsations in neutron stars expect high frequencies between

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Figure 7: Still from a Movie showing the evolution of the regular collapse model A3B2G4 ofDimmelmeier et al. [60]. The left frame contains the 2D evolution of the logarithmic density. Theupper and lower right frames display the evolutions of the gravitational wave amplitude and themaximum density, respectively. (To watch the movie, please go to the online version of this reviewarticle at http: // www. livingreviews. org/ lrr-2003-2 .)

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Figure 8: Still from a Movie showing the same as Movie 7, but for rapid collapse model A3B2G5of Dimmelmeier et al. [60]. (To watch the movie, please go to the online version of this reviewarticle at http: // www. livingreviews. org/ lrr-2003-2 .)

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Gravitational Waves from Gravitational Collapse 29

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Figure 10: Still from a Movie showing the same as Movie 7, but for rapid, differentially rotatingcollapse model A4B5G5 of Dimmelmeier et al. [60]. (To watch the movie, please go to the onlineversion of this review article at http: // www. livingreviews. org/ lrr-2003-2 .)

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1.8 – 3.6 kHz [235]. For most of their models, this increase in fGW was not accompanied by anincrease in hpk. This means that relativistic effects could decrease the detectability of GW signalsfrom some core collapses. However, the GW emission from the models of Dimmelmeier et al. couldbe detected by the first generation of ground-based interferometric detectors if the sources werefortuitously located in the Local Group of galaxies. A catalog containing the signals and spectraof the GW emission from all of their models can be found at [164].

3.4.5 Simulations of convective instabilities

Convectively driven inhomogeneities in the density distribution of the outer regions of the nascentneutron star and anisotropic neutrino emission are other sources of GW emission during the col-lapse/explosion [41, 173]. GW emission from these processes results from small-scale aspheric-ities, unlike the large-scale motions responsible for GW emission from aspherical collapse andnon-axisymmetric global instabilities. Note that Rayleigh–Taylor instabilities also induce time-dependent quadrupole moments at composition interfaces in the stellar envelope. However, theresultant GW emission is too weak to be detected because the Rayleigh–Taylor instabilities occurat very large radii [173].

Since convection was suggested as a key ingredient into the explosion, it has been postulatedthat asymmetries in the convection can produce the large proper motions observed in the pulsarpopulation [111]. Convection asymmetries can either be produced by asymmetries in the progenitorstar that grow during collapse or by instabilities in the convection itself. Burrows and Hayes [41]proposed that asymmetries in the collapse could produce the pulsar velocities. The idea behind thiswork was that asymmetries present in the star prior to collapse (in part due to convection duringsilicon and oxygen burning) will be amplified during the collapse [20, 147]. These asymmetries willthen drive asymmetries in the convection and ultimately, the supernova explosion. Burrows andHayes [41] found that not only could they produce strong motions in the nascent neutron star, butdetectable gravitational wave signals. The peak amplitude calculated was hpk ∼ 3 × 10−24, for asource located at 10 Mpc.

Fryer [80] was unable to produce the large neutron star velocities seen by Burrows and Hayes [41]even after significantly increasing the level of asymmetry in the initial star in excess of 25%. Thisdiscrepancy is now known to be due to the crude 2-dimensional model and gravity scheme usedby Burrows and Hayes [40]. However, the gravitational wave signal produced by both simulationsis comparable. Figure 11 shows the gravitational waveform from the Burrows & Hayes simulation(including separate matter and neutrino contributions). Figures 12, 13 show the matter andneutrino contributions respectively to the gravitational wave forms for the Fryer results. Thegravitational wave amplitude is dominated by the neutrino component and can exceed hpk ∼6× 10−24, for a source located at 10 Mpc in Fryer’s most extreme example.

The study by Nazin and Postnov [183] predicts a lower limit for EGW emitted during an asym-metric core collapse SN (where such asymmetries could be induced by both aspherical mass motionand neutrino emission). They assume that observed pulsar kicks are solely due to asymmetric col-lapse. They suggest that the energy associated with the kick (Mv2/2, where M and v are themass and velocity of the neutron star) can be set as a lower limit for EGW (which can be com-puted without having to know the mechanism behind the asymmetric collapse). From observedpulsar proper motions, they estimate the degree of asymmetry ε present in the collapse and thecorresponding characteristic GW amplitude (h ∝

√ε). This amplitude is 3 × 10−25 for a source

located at 10 Mpc and emitting at fGW = 1 kHz.Muller and Janka performed both 2D and 3D simulations of convective instabilities in the

proto-neutron star and hot bubble regions during the first second of the explosion phase of a TypeII SN [176]. They numerically computed the GW emission from the convection-induced aspherical

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Figure 11: The gravitational waveform (including separate matter and neutrino contributions)from the collapse simulations of Burrows and Hayes [41]. The curves plot the gravitational waveamplitude of the source as a function of time. (Figure 3 of [41]; used with permission.)

Figure 12: The gravitational waveform for matter contributions from the asymmetric collapsesimulations of Fryer et al. [87]. The curves plot the the gravitational wave amplitude of the sourceas a function of time. (Figure 3 of [87]; used with permission.)

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32 Chris L. Fryer and Kimberly C. B. New

Figure 13: The gravitational waveform for neutrino contributions from the asymmetric collapsesimulations of Fryer et al. [87]. The curves plot the product of the gravitational wave amplitude tothe source as a function of time. (Figure 8 of [87]; used with permission.)

mass motion and neutrino emission in the quadrupole approximation (for details, see Section 3 oftheir paper).

For typical iron core masses, the convectively unstable region in the proto-neutron star extendsover the inner 0.7 – 1.20M of the core mass (this corresponds to a radial range of ∼ 10 – 50 km).The convection in this region, which begins approximately 10 – 20 ms after the shock forms and maylast for ≈ 20 ms – 1 s, is caused by unstable gradients in entropy and/or lepton number resultingfrom the stalling of the prompt shock and deleptonization outside the neutrino sphere. Mullerand Janka’s simulations of convection in this region began with the 1D, non-rotating, 12 ms post-bounce model of Hillebrandt [113]. This model included general relativistic corrections that had tobe relaxed away prior to the start of the Newtonian simulations. Neutrino transport was neglectedin these runs (see Section 2.1 of [176] for justification); however, a sophisticated equation of statewas utilized. Figure 14 shows the evolution of the temperature and density distributions in the 2Dsimulation of Muller and Janka.

The peak GW amplitude resulting from convective mass motions in these simulations of theproto-neutron star was ≈ 3 × 10−24 in 2D and ≈ 2 × 10−25 in 3D, for d = 10 Mpc. Morerecent calculations get amplitudes of ≈ 10−26 in 2D [177] and ≈ 3 − 5 × 10−26 in 3D [87]. Theemitted energy was 9.8× 1044 erg in 2D and 1.3× 1042 erg in 3D. The power spectrum peaked atfrequencies of 200 – 600 Hz in 2D and 100 – 200 Hz in 3D. Such signals would not be detectable withLIGO-II. The reasons for the differences between the 2D and 3D results include smaller convectiveelements and less under- and overshooting in 3D. The relatively low angular resolution of the 3Dsimulations may have also played a role. The quadrupole gravitational wave amplitude AE2

20 fromthe 2D simulation is shown in the upper left panel of Figure 15 (see [271, 241] for expressionsrelating AE2

20 to h).Convection in the hot bubble region between the shock and neutrino sphere arises because of

an unstable entropy gradient resulting from the outward moving shock and subsequent neutrinoheating. Figure 16 shows a movie of the development of this entropy-driven convection. Thisunstable region extends over the inner mass range 1.25 – 1.40 M (corresponding to a radial rangeof ≈ 100 – 1000 km). Convection in the hot bubble begins ≈ 50 – 80 ms after shock formation

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Gravitational Waves from Gravitational Collapse 33

Figure 14: Convective instabilities inside the proto-neutron star in the 2D simulation of Muller andJanka [176]. The evolutions of the temperature (left panels) and logarithmic density (right panels)distributions are shown for the radial region 15 – 95 km. The upper and lower panels correspond totimes 12 and 21 ms, respectively, after the start of the simulation. The temperature values rangefrom 2.5 × 1010 to 1.8 × 1011 K. The values of the logarithm of the density range from 10.5 to13.3 g cm−3. The temperature and density both increase as the colors change from blue to green,yellow, and red. (Figure 7 of [176]; used with permission.)

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34 Chris L. Fryer and Kimberly C. B. New

Figure 15: Quadrupole amplitudes AE220 [cm] from convective instabilities in various models of [176].

The upper left panel is the amplitude from a 2D simulation of proto-neutron star convection. Theother three panels are amplitudes from 2D simulations of hot bubble convection. The imposedneutrino flux in the hot bubble simulations increases from the top right model through the bottomright model. (Figure 18 of [176]; used with permission.)

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Gravitational Waves from Gravitational Collapse 35

and lasts for ≈ 100 – 500 ms. Only 2D simulations were performed in this case. These runsstarted with a 25 ms post-bounce model provided by Bruenn. A simple neutrino transport schemewas used in the runs and an imposed neutrino flux was located inside the neutrino sphere. Dueto computational constraints, the computational domain did not include the entire convectivelyunstable region inside the proto-neutron star (thus this set of simulations only accurately modelsthe convection in the hot bubble region, not in the proto-neutron star).

C

B

A

Figure 16: Still from a Movie showing the isosurface of material with radial velocities of1000 km s−1 for 3 different simulation resolutions. The isosurface outlines the outward movingconvective bubbles. The open spaces mark the downflows. Note that the upwelling bubbles are largeand have very similar size scales to the two-dimensional simulations. (To watch the movie, pleasego to the online version of this review article at http: // www. livingreviews. org/ lrr-2003-2 .)

The peak GW amplitude resulting from these 2D simulations of convective mass motions in thehot bubble region was hpk ≈ 5× 10−25, for d = 10 Mpc. The emitted energy was . 2× 1042 erg.The energy spectrum peaked at frequencies of 50 – 200 Hz. As the explosion energy was increased(by increasing the imposed neutrino flux), the violent convective motions turn into simple rapidexpansion. The resultant frequencies drop to fGW ∼ 10 Hz. The amplitude of such a signal wouldbe too low to be detectable with LIGO-II.

The case for GWs from convection induced asymmetric neutrino emission has also varied withtime. Muller and Janka estimated the GW emission from the convection induced anisotropicneutrino radiation in their simulations (see [176] for details). They found that the amplitude ofthe GWs emitted can be a factor of 5 – 10 higher than the GW amplitudes resulting from convective

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36 Chris L. Fryer and Kimberly C. B. New

mass motion. Muller et al. (2004) [177] argue now that the GWs produced by asymmetric neutrinoemission is less than that of the convective motions.

Our understanding of the convective engine is evolving with time. Scheck et al. [212] found thatthe convective cells could merge with time, producing a single lobe convective instability, fulfillingthe prediction by Herant [111]. Blondin et al. [26] argue that a standing shock instability coulddevelop to drive low-mode convection and Burrows et al. [43] argue that it is the shocks producedin this convection that truly drives the supernova explosion. This convection can drive oscillationsin the neutron star which may also be a source for GWs (see Fig. 17).

Figure 17: Still from a Movie showing the oscillation of the proto-neutron star caused by acousticinstabilities in the convective region above the shock. (To watch the movie, please go to the onlineversion of this review article at http: // www. livingreviews. org/ lrr-2003-2 .)

3.5 Going further

The background of GW emission from a population of core collapse SNe at cosmological distancesmay be detectable by LIGO-II, according to Ferrari, Matarrese, and Schneider [69]. They deter-mined the SN rate as a function of redshift using observations to determine the evolution of the starformation rate. Only collapses that lead to black hole formation were considered. This simplifiedthe study because the GW emission from such collapses is generally a function of just the black holemass and angular momentum. They found that the stochastic background from these sources isnot continuous and suggest that this could be used to optimize detection strategies. The maximumGW spectral strain amplitude they computed was in the range 10−28 – 10−27 Hz, at frequenciesof a few times 102 Hz. Such a signal may be detected by a pair of LIGO-II detectors. Buonannoet al. [38] have recently redone such a study with all the current results on supernovae, arguingthat the GW background would be detectable by second-generation (e.g., Big Bang Observatory)space-based detectors, noting that Pop III (black-hole forming stars) could well dominate the totalbackground.

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Gravitational Waves from Gravitational Collapse 37

4 Collapsar Models

4.1 Collapse scenario

Without winds, stars above ∼ 40 – 50 M will not produce supernova explosions, but collapsedirectly to black holes [78]. Stars above ∼ 20 – 25 M are believed to produce only weak supernovaexplosions. These explosions are unable to eject all of the star’s mantle, and will ultimately alsocollapse to black holes as this mantle falls back onto the core [78]. If these stars are rotating, theywill develop accretion disks around the newly formed black hole, opening up new ways to produceboth GWs and baryonic explosions.

These black hole accretion disks formed by stellar collapse are currently the favored enginebehind long-duration gamma-ray bursts [262, 160]. It is believed that either neutrinos rising fromthe disk annihilate and drive an explosion or magnetic fields wound up in the disk can producejets that then evolve into the highly relativistic outflows observed in the GRBs [195]. Relativisticand magnetic field effects are far more important in these calculations than those for normalstellar collapse models. And yet, most simulations of GRBs to date do not include the detailedmicrophysics, magnetic fields (although see Proga and collaborators [196]), and relativity at thesame level of sophistication as the simulations of core-collapse supernovae. Even so, this topic hasgarnered some interest in the GW community and this interest will only grow as the simulationsbecome more accurate.

4.2 Formation rate

The fraction of massive stars collapsing to black holes can be estimated theoretically and liesbetween 5 – 40% [89]. But it is much more difficult to determine the fraction of these black-holeforming stars that also have enough rotation to produce a GRB. Likewise, the wide-field monitoringof the night sky of GRBs produces an accurate observed rate of GRBs, but this rate must becorrected by beaming factors to get a true long-duration GRB rate. Observations of supernovaremnants have allowed GRB observers to place some constraints on the GRB rate relative to theType Ib/c supernova rate: RateGRB/RateSNIb/c < 3% [22]. This predicts a Galactic rate that isbelow ∼ 10−4 yr−1 and the true value is likely to lie between ∼ 10−5 – 10−4 yr−1. Correlatingthe observed GRB rate to the rate of black hole accretion disks formed in stellar collapse is notstraightforward. First, it may be that long-duration GRBs are not formed from stellar collapse,but from stellar mergers [93, 94]. If this is the case, then we do not expect any GW signal fromthe collapse itself, but any disk instabilities may still produce GW signals. Second, there may bea number of black hole accretion disk systems produced that do not produce gamma-ray burstsbecause they can not achieve the high relativistic flows, increasing the total rate of GW sources.

4.3 GW emission mechanisms

If the collapsed remnant is a black hole, GWs will be radiated as the infall of the remaining stellarmatter distorts the black hole’s geometry. This “ringdown phase” will end when gravitationalradiation has dissipated all of the black hole’s accretion-induced distortion. Zanotti, Rezzolla,and Font [269] have suggested that the torus of matter surrounding the black hole may be aneven stronger source of GWs than the collapse itself (see also [140, 141, 251, 250]). Kobayashi& Meszaros (2003) have further found that the GW signal will be polarized and this polarizationmay be detectable.

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38 Chris L. Fryer and Kimberly C. B. New

4.4 Numerical Predictions of GW emission Mechanisms

The first fully general relativistic investigations of stellar core collapse were Nakamura’s 2D simu-lations of rotating collapse [179, 180]. However, because of the limits of his numerical formalismand computational resources, he was unable to compute the emitted gravitational radiation (theenergy of this emission is quite small compared to the rest mass energy and thus was difficult toextract numerically). The results of this work indicate that collapse does not lead to black holeformation if the parameter q = J/M2 exceeds unity (here J and M are the angular momentumand gravitational mass of the remnant).

Fully general relativistic collapse simulations (i.e., without the conformally flat approximation)have also been performed by Shibata [222]. He used an axisymmetric code that solves the Einsteinequations in Cartesian coordinates and the hydrodynamics equations in cylindrical coordinates.The use of the Cartesian grid eliminates the presence of singularities and allows for stable, long-duration axisymmetric simulations [4]. The focus of this work was the effect of rotation on thecriteria for prompt black hole formation. Shibata found that if the parameter q = J/M2 is lessthan 0.5, black hole formation occurred for rest masses slightly greater than the maximum mass ofspherical stars. However, for 0.5 < q < 1, the maximum stable rest mass is increased by ∼ 70 – 80%.The results are only weakly dependent on the initial rotation profile. More recent results suggestthat this limit can be eased for differentially rotating massive stellar cores, and systems with spinparameter q as high as 2.5 may collapse to form black holes [217]. Shibata did not compute theGW emission in his collapse simulations, but in a recent study using axisymmetric calculations,GW signals have been calculataed focusing on this collapse phase [267].

Duez et al. [62] found that if a black hole does form, but the disk is spinning rapidly, that thedisk will fragment and its subsequent accretion will be in spurts, causing a “splash” onto the blackhole, producing ringing and GW emission. Their result implies very strong gravitational waveamplitudes & 10−21 at distances of 10 Mpc. Black hole ringing was also estimated by FHH, wherethey too assumed discrete accretion events. They found that even with very optimistic accretionscenarios, that such radiation will be of very low amplitude and beyond the upper frequency reachof LIGO-II (see [86] for details).

The new general relativistic hydrodynamics simulations of Zanotti, Rezzolla, and Font [269]suggest that a torus of neutron star matter surrounding a black hole remnant may be a strongersource of GWs than the collapse itself. They used a high resolution shock-capturing hydrodynamicsmethod in conjunction with a static (Schwarzschild) spacetime to follow the evolution of “toroidalneutron stars”. Their results indicate that if a toroidal neutron star (with constant specific angularmomentum) is perturbed, it could undergo regular oscillations. They estimate that the resultingGW emission would have a characteristic amplitude hc ranging from 6 × 10−24 – 5 × 10−23, forratios of torus mass to black hole mass in the range 0.1 – 0.5. (These amplitude values are likelyunderestimated because the simulations of Zanotti et al. are axisymmetric.) The correspondingfrequency of emission is fGW ∼ 200 Hz. The values of hc and fGW quoted here are for a sourcelocated at 10 Mpc. This emission would be just outside the range of LIGO-II (see Figure 2).Further numerical investigations, which study tori with non-constant angular momenta and includethe effects of self-gravity and black hole rotation, are needed to confirm these predictions. Moviesfrom the simulations of Zanotti et al. can be viewed at [200].

Magnetized tori around rapidly spinning black holes (formed via either core collapse or neutronstar-black hole coalescence) have recently been examined in the theoretical study of van Puttenand Levinson [251]. They find that such a torus–black hole system can exist in a suspended stateof accretion if the ratio of poloidal magnetic field energy to kinetic energy EB/Ek is less than 0.1.They estimate that ∼ 10% of the spin energy of the black hole will be converted to gravitationalradiation energy through multipole mass moment instabilities that develop in the torus. If amagnetized torus–black hole system located at 10 Mpc is observed for 2 × 104 rotation periods,

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Gravitational Waves from Gravitational Collapse 39

the characteristic amplitude of the GW emission is ∼ 6 × 10−20. It is possible that this emissioncould take place at several frequencies. Observations of x-ray lines from gamma-ray bursts (whichare possibly produced by these types of systems) could constrain these frequencies by providinginformation regarding the angular velocities of the tori: preliminary estimates from observationssuggest fGW ∼ 500 Hz, placing the radiation into a range detectable by LIGO-I [251].

4.5 Going further

As we noted in Section 3.5, the background of GW emission is likely dominated by the populationof stars that collapse to black holes. We refer the reader back to this section on progress in thisarena.

Baumgarte and collaborators have studied the neutrino signal from the collapse of a hot neutronstar down to a black hole [18, 19, 15]. This collapse phase has a specific neutrino signal, that whencompared with the gravitational wave observations, can tell us a lot about the behavior of matterat nuclear densities.

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40 Chris L. Fryer and Kimberly C. B. New

5 Collapse of Population III Stars

5.1 Collapse scenario

The first generation of stars to form in the early universe are known as Population III stars (formedat redshifts z & 5). Theoretical and computational evidence suggests that Population III stars mayhave had masses & 100 M [2, 1, 96]. Since these massive stars contained no metals, it was possiblefor them to form directly and then evolve with very low stellar winds and thus very little massloss. If the mass of a nonrotating Population III star is & 260 M, its fate is to collapse directly toa black hole at the end of its life [96]. If rotational support prevents the star from direct collapseto a black hole, explosive thermonuclear burning will cause the star to undergo a giant hypernovaexplosion. Prior to black hole formation, the rotating, collapsed core will have a mass of 50 – 70Mand a radius of 1000 – 2000 km. Note that because these massive stars evolve so quickly (in a fewmillion years [10]), the events associated with their deaths will take place at roughly the redshiftsof their births.

5.2 Formation rate

The formation rate of Population III stars can be indirectly estimated from the re-ionizationfraction of the early universe, which was re-ionized by light from these stars [117, 47]. Usingestimates of the ultraviolet light emission from Population III stars, their ionization efficiency, andthe re-ionization fraction of the early universe, one can determine that about 0.01% – 1% of theuniverse’s baryonic matter was found in these very massive stars. This corresponds to ∼ 104 – 107

Population III stars in a 1011 M galaxy and thus a collapse rate that is . 10−3 yr−1. Thus,a reasonable occurrence rate can be found for an observation (luminosity) distance of ∼ 50 Gpc(which corresponds to a redshift of z = 5, in the cosmology used by [86]). However, uncertaintiesin the assumptions make this formation rate uncertain by a few orders of magnitude [1, 96, 2, 86].Indeed, the latest results suggest that the initial mass function begins to fall off dramaticallyabove 100 M and the number of stars fomred with masses above 25 M may be many orders ofmagnitude below this rate[189]. Instead, Population III stars may be dominated by the collapse ofstars in the 20 – 100 M. These stars may also collapse ultimately to black holes. For discussionof these objects, we refer the reader back to Section 4.

5.3 GW emission mechanisms

The GW emission mechanisms related to the collapse of Population III stars are a subset of thosediscussed in the sections on AIC and SNe/collapsars. These include aspherical collapse, globalrotational and fragmentation instabilities that may arise during the collapse/explosion and in thecollapse remnant (prior to black hole formation), and the “ring-down” of the remnant black hole.

5.4 Numerical predictions of GW emission

The GW emission from the collapse of Population III stars has been investigated by Fryer and col-laborators (Fryer, Woosley, and Heger [96], FHH [86], and Fryer, Holz, Hughes, and Warren [88]).The collapse simulations of Fryer, Woosley, and Heger again started with rotating collapse pro-genitors that had been evolved with a stellar evolution code [107]. The initial models used by theevolution code were in rigid rotation with a surface ratio of centrifugal to gravitational forces of20% (this ratio is seen in current observations of O stars).

The results of Fryer, Woosley, and Heger suggest that the collapse remnant (prior to black holeformation) is susceptible to the development of a secular bar-mode instability. However, at z > 5,the GW emission would be redshifted out of LIGO-II’s frequency range. At z = 5, hpk = 8×10−23,

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Gravitational Waves from Gravitational Collapse 41

with a corresponding frequency of 10 Hz [86, 88]. Even if such a signal persists for a hundredcycles, it probably would be undetectable by LIGO-II. Note that these signal strengths are ordersof magnitude lower than the qualitative estimates of signal strength given in Carr, Bond, andArnett [47].

LIGO-II may be able to detect the GW emission from binary clumps formed via a fragmentationinstability. If such a signal is emitted at z = 5 and persists for 10 cycles, h would be ∼ 10−22, overa frequency range of 10 – 100 Hz [86, 88]. The likelihood of the development of a fragmentationinstability is diminished by the fact that the off-center density maxima present in the simulationsof Fryer, Woosley, and Heger are not very pronounced.

The “ring-down” of the black hole remnant will likely be strong because Fryer, Woosley, andHeger observe a high accretion rate after collapse. FHH estimate that for a source located atz = 20, the GWs would be redshifted out of LIGO-II’s bandwidth. However, for a source at z = 5,hpk ∼ 6× 10−23 and the frequency range is 20 – 70 Hz. This signal may be marginally detectablewith LIGO-II (see Figure 2).

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42 Chris L. Fryer and Kimberly C. B. New

6 Collapse of Supermassive Stars

6.1 Collapse scenario

There is a large body of observational evidence that supermassive black holes (SMBHs, M &106 M) exist in the centers of many, if not most galaxies (see, e.g., the reviews of Rees [199]and Macchetto [159]). The masses of SMBHs in the centers of more than 45 galaxies have beenestimated from observations [68] and there are more than 30 galaxies in which the presence of aSMBH has been confirmed [142].

One of the possible formation mechanisms for SMBHs involves the gravitational collapse ofsupermassive stars (SMSs). The timescale for this formation channel is short enough to accountfor the presence of SMBHs at redshifts z > 6 [129]. Supermassive stars may contract directly out ofthe primordial gas, if radiation and/or magnetic field pressure prevent fragmentation [101, 64, 100,157, 33, 1]. Alternatively, they may build up from fragments of stellar collisions in clusters [211, 21].Supermassive stars are radiation dominated, isentropic and convective [221, 270, 157]. Thus, theyare well represented by an n = 3 polytrope. If the star’s mass exceeds 106 M, nuclear burningand electron/positron annihilation are not important.

After formation, an SMS will evolve through a phase of quasistationary cooling and contraction.If the SMS is rotating when it forms, conservation of angular momentum requires that it spins upas it contracts. There are two possible evolutionary regimes for a cooling SMS. The path takenby an SMS depends on the strength of its viscosity and magnetic fields and on the nature of itsangular momentum distribution.

In the first regime, viscosity or magnetic fields are strong enough to enforce uniform rotationthroughout the star as it contracts. Baumgarte and Shapiro [16] have studied the evolution ofa uniformly rotating SMS up to the onset of relativistic instability. They demonstrated thata uniformly rotating, cooling SMS will eventually spin up to its mass shedding limit. The massshedding limit is encountered when matter at the star’s equator rotates with the Keplerian velocity.The limit can be represented as βshed = (T/|W |)shed. In this case, βshed = 9 × 10−3. The starwill then evolve along a mass shedding sequence, losing both mass and angular momentum. It willeventually contract to the onset of relativistic instability [123, 50, 51, 221, 129].

Baumgarte and Shapiro used both a second-order, post-Newtonian approximation and a fullygeneral relativistic numerical code to determine that the onset of relativistic instability occurs ata ratio of R/M ∼ 450, where R is the star’s radius and G = c = 1 in the remainder of this section.Note that a second-order, post-Newtonian approximation was needed because rotation stabilizesthe destabilizing role of nonlinear gravity at the first post-Newtonian level. If the mass of the starexceeds 106 M, the star will then collapse and possibly form a SMBH. If the star is less massive,nuclear reactions may lead to explosion instead of collapse.

The major result of Baumgarte and Shapiro’s work is that the universal values of the followingratios exist for the critical configuration at the onset of relativistic instability: T/|W |, R/M , andJ/M2. These ratios are completely independent of the mass of the star or its prior evolution. Be-cause uniformly rotating SMSs will begin to collapse from a universal configuration, the subsequentcollapse and the resulting gravitational waveform will be unique.

In the opposite evolutionary regime, neither viscosity nor magnetic fields are strong enough toenforce uniform rotation throughout the cooling SMS as it contracts. In this case, it has been shownthat the angular momentum distribution is conserved on cylinders during contraction [27]. Becauseviscosity and magnetic fields are weak, there is no means of redistributing angular momentum inthe star. So, even if the star starts out rotating uniformly, it cannot remain so.

The star will then rotate differentially as it cools and contracts. In this case, the subsequentevolution depends on the star’s initial angular momentum distribution, which is largely unknown.One possible outcome is that the star will spin up to mass-shedding (at a different value of βshed

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Gravitational Waves from Gravitational Collapse 43

than a uniformly rotating star) and then follow an evolutionary path that may be similar to thatdescribed by Baumgarte and Shapiro [16]. The alternative outcome is that the star will encounterthe dynamical bar instability prior to reaching the mass-shedding limit. New and Shapiro [185, 186]have demonstrated that a bar-mode phase is likely to be encountered by differentially rotating SMSswith a wide range of initial angular momentum distributions. This mode will transport mass andangular momentum outward and thus may hasten the onset of collapse.

6.2 Formation rate

An estimate of the rate of the collapse of SMSs can be derived from the quasar luminosity function.Haehnelt [99] has used the quasar luminosity function to compute the rate of GW bursts fromsupermassive black holes, assuming that each quasar emits one such burst during its lifetime (andthat each quasar is a supermassive black hole). If it is assumed that each of these bursts is due tothe formation of a supermassive black hole via the collapse of a SMS, then Haehnelt’s rate estimatescan be used as estimates of the rate of SMS collapse. This rate is likely an overestimate of the SMScollapse rate because many SMBHs may have been formed via merger. Haehnelt predicts that theintegrated event rate through redshift z = 4.5 ranges from ∼ 10−6 yr−1 for M = 108 M objectsto ∼ 1 yr−1 for M = 106 M objects. Thus, as in the case of Population III stars, a reasonableoccurrence rate can be determined for an observation (luminosity) distance of 50 Gpc.

6.3 GW emission mechanisms

The GW emission mechanisms related to the collapse of SMSs are a subset of those discussedin the sections on AIC, SNe/collapsars, and Population III stellar collapse. These include (i)aspherical collapse, (ii) global rotational and fragmentation instabilities that may arise duringthe collapse/explosion and in the collapsed remnant (prior to black hole formation), and (iii) the“ring-down” of the remnant black hole.

6.4 Numerical predictions of GW emission

The outcome of SMS collapse can be determined only with numerical, relativistic 3D hydrodynam-ics simulations.

Until recently, such simulations had been published only for nearly spherical collapse. Thespherical simulations of Shapiro and Teukolsky [220] produced collapse evolutions that were nearlyhomologous. In this case, the collapse time τcoll is roughly the free-fall time at the horizon

τcoll =(

R3

4πM

)1/2

= 14 s(

M

106 M

)−1

. (7)

The peak GW frequency fGW = τ−1coll is then 10−2 Hz, if the mass of the star is 106 M. This is in

the middle of LISA’s frequency band of 10−4 – 1 Hz [243, 76].The amplitude h of this burst signal can be roughly estimated in terms of the star’s quadrupole

moment

h ≤ ε2M2

Rd

≤ ε · 1× 10−18

(M

106 M

) (d

50 Gpc

)−1

. (8)

Here d is the distance to the star and ε ∼ T/|W | is a measure of the star’s deviation from sphericalsymmetry. In this case, ε will be much less than one near the horizon, since the collapse is nearlyspherical.

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44 Chris L. Fryer and Kimberly C. B. New

There are two possible aspherical collapse outcomes that can be discussed. The first outcomeis direct collapse to a SMBH. In this case, ε will be on the order of one near the horizon. Thus,according to Equation 8, the peak amplitude of the GW burst signal will be

hpk ∼ 1× 10−18

(M

106 M

) (d

50 Gpc

)−1

. (9)

Alternatively, the star may encounter the dynamical bar mode instability prior to completecollapse. Baumgarte and Shapiro [16] have estimated that a uniformly rotating SMS will reachβ ∼ 0.27 when R/M = 15. The frequency of the quasiperiodic gravitational radiation emitted bythe bar can be estimated in terms of its rotation frequency to be

fGW = 2fbar ∼ 2(

GM

R3

)1/2

= 2× 10−3 Hz(

M

106 M

)−1

, (10)

when R/M = 15. The corresponding hpk, again estimated in terms of the star’s quadrupolemoment, is

hpk ≤2M2

Rd

≤ 1× 10−19

(M

106 M

) (d

50 Gpc

)−1

. (11)

The LISA sensitivity curve is shown in Figure 18 (see [119] for details on the computation of thiscurve; a mission time of 3 years has been assumed). The GW signal from this dynamical bar-modecould be detected with LISA.

Shibata and Shapiro [228] have published a fully general relativistic, axisymmetric simulation ofthe collapse of a rapidly, rigidly rotating SMS. They found that the collapse remained homologousduring the early part of the evolution. An apparent horizon does appear in their simulation,indicating the formation of a black hole. Because of the symmetry condition used in their run,non-axisymmetric instabilities were unable to develop.

The collapse of a uniformly rotating SMS has been investigated with post-Newtonian hydro-dynamics, in 3+1 dimensions, by Saijo, Baumgarte, Shapiro, and Shibata [208]. Their numericalscheme used a post-Newtonian approximation to the Einstein equations, but solved the fully rela-tivistic hydrodynamics equations. Their initial model was an n = 3 polytrope.

The results of Saijo et al. (confirmed in conformally flat simulations [207]) indicate that thecollapse of a uniformly rotating SMS is coherent (i.e., no fragmentation instability develops). Thecollapse evolution of density contours from their model is shown in Figure 19. Although the work ofBaumgarte and Shapiro [16] suggests that a bar instability should develop prior to BH formation,no bar development was observed by Saijo et al. They use the quadrupole approximation toestimate a mean GW amplitude from the collapse itself: h = 4× 10−21, for a 106 M star locatedat a distance of 50 Gpc. Their estimate for fGW at the time of BH formation is 3× 10−3 Hz. Thissignal would be detectable with LISA (see Figure 18).

Saijo et al. also consider the GW emission from the ringdown of the black hole remnant. For thel = m = 2 quasi-normal mode of a Kerr black hole with a/M = 0.9, they estimate the characteristicamplitude of emission to be h ∼ 1.2 × 10−20[(4EGW/M)/10−4]1/2 at fGW ∼ 2 × 10−2 Hz for anM = 106 M source located at a luminosity distance of 50 Gpc (see [149, 242, 229] for details).Here, 4EGW/M is the radiated energy efficiency and may be . 7 × 10−4 [233]. This GW signalis within LISA’s range of sensitivity (see Figure 18).

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Gravitational Waves from Gravitational Collapse 45

Figure 18: A comparison between the GW amplitude h(f) for various sources and the LISA noisecurve. See the text for details regarding the computations of h. The SMS sources are assumed tobe located at a luminosity distance of 50 Gpc. The bar-mode source is a dynamical bar-mode.

7 Summary

It is hoped that as gravitational collapse simulations become more sophisticated, the historicallywidely varying estimates of the magnitude of GW emission from collapse may start to converge.Steady progress in this field has been made in the last decade. Some researchers have begunto use progenitor models produced with stellar evolution codes, which thus have more realisticangular momentum profiles, as starting points for collapse simulations [86]. This reduces the needfor collapse studies that include large surveys of the angular momentum parameter space. Otherprogress made in the numerical study of collapse includes the use of realistic equations of state [86],advanced neutrino transport and interaction schemes [130, 150, 197, 240], and the performance of3D Newtonian [198, 35, 91, 92] and improved methods for general relativistic simulations [222, 61].

There is still much work to be done toward the goal of self-consistent, 3D general relativisticcollapse simulations. The most rapidly rotating progenitors (with possibly the strongest GW sig-nals) may be produced only in specific binary systems [95, 84, 265]. If the GW signal is dominatedby convection, then it is critical that scientists actually understand the true supernova mecha-nism. Accurate progenitor modelling and collapse simulations must include the effects of magneticfields, as they can significantly alter the amount of angular momentum and differential rotationpresent in collapsing stars. Many of the more advanced studies, which include proper microphysicstreatment and/or general relativistic effects, have been limited to axisymmetry. Full 3D simu-lations are necessary to compute the characteristics of the GW emission from non-axisymmetriccollapse phenomena. Furthermore, simulations that follow both the collapse and the evolutionof the collapsed remnant are necessary to consistently predict GW emission. One benefit of longduration simulations is that they will facilitate the investigation of the effects of the envelope onany instabilities that develop in the collapsing core or remnant. Of course, lengthy 3D simulationsare computationally intensive. This burden may be reduced by the use of advanced numerical

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46 Chris L. Fryer and Kimberly C. B. New

-50 -25 0 25 500

1020

z /M

-30 -20 -10 0 10 20 30x /M

05

1015

z /M

-600 -400 -200 0 200 400 6000

200

400

z /M

-30 -20 -10 0 10 20 3005

1015

z/M

0

00

00 (a)

(b)

(c)

(d)

Figure 19: Meridional plane density contours from the SMS collapse simulation of Saijo, Baum-garte, Shapiro, and Shibata [208]. The contour lines denote densities ρ = ρc × d(1−i/16), where ρc

is the central density. The frames are plotted at (t/tD, ρc, d)=(a)(5.0628 × 10−4, 8.254 × 10−9,10−7), (b)(2.50259, 1.225 × 10−4, 10−5), (c)(2.05360, 8.328 × 10−3, 5.585 × 10−7), (d)(2.50405,3.425 × 10−2, 1.357 × 10−7), respectively. Here t, tD, and M0 are the time, dynamical time(=

√R3

e/M , where Re is the initial equatorial radius and M is the mass), and rest mass. (Fig-ure 15 of [208]; used with permission.)

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Gravitational Waves from Gravitational Collapse 47

techniques, including adaptive mesh refinement and parallel algorithms.The current numerical simulations of gravitational collapse indicate that interferometric obser-

vatories could detect GWs emitted by some collapse phenomena. LIGO-I may be able to detectGWs from secular bar-mode instabilities in core collapse SNe [146] and magnetized tori surround-ing black hole collapse remnants [251]. LIGO-II could observe GWs from dynamical bar-modeinstabilities in AIC [155] and core collapse SNe [86]. LISA should be able to detect the collapse(and any bar-mode instabilities that develop during the collapse) of SMSs [16] and the ringdownof black hole remnants of collapsed Population III stars [86] and SMSs [208]. These observationswill provide unique information about gravitational collapse, the supernova and gamma-ray burstexplosion mechanisms, and their associated progenitors and remnants.

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48 Chris L. Fryer and Kimberly C. B. New

8 Acknowledgements

It is a pleasure to thank Paul Bradley, Adam Burrows, Harald Dimmelmeier, Alex Heger, ScottHughes, Hui Li, Ewald Muller, Ken Nomoto, Shangli Ou, and Stuart Shapiro for helpful conversa-tions and/or permission to reprint figures/movies from their published works. We also gratefullyacknowledge Ewald Muller and a second referee for their beneficial reviews of this article’s firstversion. This work was performed under the auspices of the U.S. Department of Energy by theLos Alamos National Laboratory under contract W-7405-ENG-36.

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